Properties

Label 4-10e2-1.1-c25e2-0-1
Degree 44
Conductor 100100
Sign 11
Analytic cond. 1568.131568.13
Root an. cond. 6.292826.29282
Motivic weight 2525
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.19e3·2-s − 9.70e4·3-s + 5.03e7·4-s − 4.88e8·5-s − 7.95e8·6-s − 1.61e10·7-s + 2.74e11·8-s − 1.24e12·9-s − 4.00e12·10-s + 1.44e13·11-s − 4.88e12·12-s + 9.34e13·13-s − 1.32e14·14-s + 4.74e13·15-s + 1.40e15·16-s + 4.99e14·17-s − 1.01e16·18-s − 3.83e15·19-s − 2.45e16·20-s + 1.56e15·21-s + 1.18e17·22-s + 1.69e17·23-s − 2.66e16·24-s + 1.78e17·25-s + 7.65e17·26-s + 1.59e17·27-s − 8.12e17·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.105·3-s + 3/2·4-s − 0.894·5-s − 0.149·6-s − 0.440·7-s + 1.41·8-s − 1.46·9-s − 1.26·10-s + 1.38·11-s − 0.158·12-s + 1.11·13-s − 0.623·14-s + 0.0943·15-s + 5/4·16-s + 0.207·17-s − 2.07·18-s − 0.397·19-s − 1.34·20-s + 0.0464·21-s + 1.95·22-s + 1.61·23-s − 0.149·24-s + 3/5·25-s + 1.57·26-s + 0.205·27-s − 0.660·28-s + ⋯

Functional equation

Λ(s)=(100s/2ΓC(s)2L(s)=(Λ(26s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}
Λ(s)=(100s/2ΓC(s+25/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 100100    =    22522^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 1568.131568.13
Root analytic conductor: 6.292826.29282
Motivic weight: 2525
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 100, ( :25/2,25/2), 1)(4,\ 100,\ (\ :25/2, 25/2),\ 1)

Particular Values

L(13)L(13) \approx 6.9585344746.958534474
L(12)L(\frac12) \approx 6.9585344746.958534474
L(272)L(\frac{27}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1 (1p12T)2 ( 1 - p^{12} T )^{2}
5C1C_1 (1+p12T)2 ( 1 + p^{12} T )^{2}
good3D4D_{4} 1+3596p3T+1717509338p6T2+3596p28T3+p50T4 1 + 3596 p^{3} T + 1717509338 p^{6} T^{2} + 3596 p^{28} T^{3} + p^{50} T^{4}
7D4D_{4} 1+16137160124T7807648962814970958p2T2+16137160124p25T3+p50T4 1 + 16137160124 T - 7807648962814970958 p^{2} T^{2} + 16137160124 p^{25} T^{3} + p^{50} T^{4}
11D4D_{4} 114416600801344T+ 1 - 14416600801344 T + 18 ⁣ ⁣6618\!\cdots\!66p2T214416600801344p25T3+p50T4 p^{2} T^{2} - 14416600801344 p^{25} T^{3} + p^{50} T^{4}
13D4D_{4} 193404548866628T+ 1 - 93404548866628 T + 11 ⁣ ⁣1411\!\cdots\!14pT293404548866628p25T3+p50T4 p T^{2} - 93404548866628 p^{25} T^{3} + p^{50} T^{4}
17D4D_{4} 129368640128308pT+ 1 - 29368640128308 p T + 34 ⁣ ⁣4234\!\cdots\!42p2T229368640128308p26T3+p50T4 p^{2} T^{2} - 29368640128308 p^{26} T^{3} + p^{50} T^{4}
19D4D_{4} 1+201589715363480pT+ 1 + 201589715363480 p T + 39 ⁣ ⁣1839\!\cdots\!18p2T2+201589715363480p26T3+p50T4 p^{2} T^{2} + 201589715363480 p^{26} T^{3} + p^{50} T^{4}
23D4D_{4} 1169271186050723788T+ 1 - 169271186050723788 T + 11 ⁣ ⁣1411\!\cdots\!14pT2169271186050723788p25T3+p50T4 p T^{2} - 169271186050723788 p^{25} T^{3} + p^{50} T^{4}
29D4D_{4} 1569443622853716940T+ 1 - 569443622853716940 T + 69 ⁣ ⁣9869\!\cdots\!98T2569443622853716940p25T3+p50T4 T^{2} - 569443622853716940 p^{25} T^{3} + p^{50} T^{4}
31D4D_{4} 13982769593868414584T+ 1 - 3982769593868414584 T + 41 ⁣ ⁣6641\!\cdots\!66T23982769593868414584p25T3+p50T4 T^{2} - 3982769593868414584 p^{25} T^{3} + p^{50} T^{4}
37D4D_{4} 128241444663789328196T 1 - 28241444663789328196 T - 14 ⁣ ⁣8214\!\cdots\!82T228241444663789328196p25T3+p50T4 T^{2} - 28241444663789328196 p^{25} T^{3} + p^{50} T^{4}
41D4D_{4} 136676428854020432524T+ 1 - 36676428854020432524 T + 35 ⁣ ⁣4635\!\cdots\!46T236676428854020432524p25T3+p50T4 T^{2} - 36676428854020432524 p^{25} T^{3} + p^{50} T^{4}
43D4D_{4} 1 1 - 90 ⁣ ⁣2890\!\cdots\!28T+ T + 34 ⁣ ⁣8234\!\cdots\!82T2 T^{2} - 90 ⁣ ⁣2890\!\cdots\!28p25T3+p50T4 p^{25} T^{3} + p^{50} T^{4}
47D4D_{4} 1 1 - 95 ⁣ ⁣9695\!\cdots\!96T+ T + 42 ⁣ ⁣1842\!\cdots\!18T2 T^{2} - 95 ⁣ ⁣9695\!\cdots\!96p25T3+p50T4 p^{25} T^{3} + p^{50} T^{4}
53D4D_{4} 1 1 - 27 ⁣ ⁣8827\!\cdots\!88T+ T + 16 ⁣ ⁣2216\!\cdots\!22T2 T^{2} - 27 ⁣ ⁣8827\!\cdots\!88p25T3+p50T4 p^{25} T^{3} + p^{50} T^{4}
59D4D_{4} 1 1 - 45 ⁣ ⁣8045\!\cdots\!80T T - 18 ⁣ ⁣0218\!\cdots\!02T2 T^{2} - 45 ⁣ ⁣8045\!\cdots\!80p25T3+p50T4 p^{25} T^{3} + p^{50} T^{4}
61D4D_{4} 1+ 1 + 26 ⁣ ⁣3626\!\cdots\!36T+ T + 93 ⁣ ⁣2693\!\cdots\!26T2+ T^{2} + 26 ⁣ ⁣3626\!\cdots\!36p25T3+p50T4 p^{25} T^{3} + p^{50} T^{4}
67D4D_{4} 1+ 1 + 14 ⁣ ⁣2414\!\cdots\!24T+ T + 65 ⁣ ⁣5865\!\cdots\!58T2+ T^{2} + 14 ⁣ ⁣2414\!\cdots\!24p25T3+p50T4 p^{25} T^{3} + p^{50} T^{4}
71D4D_{4} 1+ 1 + 17 ⁣ ⁣3617\!\cdots\!36T+ T + 34 ⁣ ⁣2634\!\cdots\!26T2+ T^{2} + 17 ⁣ ⁣3617\!\cdots\!36p25T3+p50T4 p^{25} T^{3} + p^{50} T^{4}
73D4D_{4} 1+ 1 + 34 ⁣ ⁣3234\!\cdots\!32T+ T + 83 ⁣ ⁣4283\!\cdots\!42T2+ T^{2} + 34 ⁣ ⁣3234\!\cdots\!32p25T3+p50T4 p^{25} T^{3} + p^{50} T^{4}
79D4D_{4} 1+ 1 + 92 ⁣ ⁣0092\!\cdots\!00T+ T + 76 ⁣ ⁣9876\!\cdots\!98T2+ T^{2} + 92 ⁣ ⁣0092\!\cdots\!00p25T3+p50T4 p^{25} T^{3} + p^{50} T^{4}
83D4D_{4} 1 1 - 68 ⁣ ⁣4868\!\cdots\!48T T - 11 ⁣ ⁣3811\!\cdots\!38T2 T^{2} - 68 ⁣ ⁣4868\!\cdots\!48p25T3+p50T4 p^{25} T^{3} + p^{50} T^{4}
89D4D_{4} 1 1 - 38 ⁣ ⁣8038\!\cdots\!80T+ T + 14 ⁣ ⁣9814\!\cdots\!98T2 T^{2} - 38 ⁣ ⁣8038\!\cdots\!80p25T3+p50T4 p^{25} T^{3} + p^{50} T^{4}
97D4D_{4} 1+ 1 + 16 ⁣ ⁣6416\!\cdots\!64T+ T + 13 ⁣ ⁣3813\!\cdots\!38T2+ T^{2} + 16 ⁣ ⁣6416\!\cdots\!64p25T3+p50T4 p^{25} T^{3} + p^{50} T^{4}
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   L(s)=p j=14(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−14.87198458028148213054408099432, −14.56420477160205827152756985243, −13.79238779399364083481388158639, −13.17722026568140347982586287076, −12.10701551749495798808679016525, −12.00045798559841793275080789727, −10.92490040001171314166845916862, −10.90507687333009033194744043577, −9.150923791724951314718736346031, −8.674934187930878223447820595810, −7.59939937718354106091567344807, −6.77611697309666041293193480564, −6.03026217565657804221099389387, −5.61075416557292730241010202054, −4.20830786797002826222917943676, −4.13850937697760931376022353106, −2.94693380386204346244762430629, −2.79622925444236206197409075218, −1.23991409486712841662795472706, −0.67250797503421422148198550273, 0.67250797503421422148198550273, 1.23991409486712841662795472706, 2.79622925444236206197409075218, 2.94693380386204346244762430629, 4.13850937697760931376022353106, 4.20830786797002826222917943676, 5.61075416557292730241010202054, 6.03026217565657804221099389387, 6.77611697309666041293193480564, 7.59939937718354106091567344807, 8.674934187930878223447820595810, 9.150923791724951314718736346031, 10.90507687333009033194744043577, 10.92490040001171314166845916862, 12.00045798559841793275080789727, 12.10701551749495798808679016525, 13.17722026568140347982586287076, 13.79238779399364083481388158639, 14.56420477160205827152756985243, 14.87198458028148213054408099432

Graph of the ZZ-function along the critical line