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Properties

Label 4-315e2-1.1-c3e2-0-6
Degree 44
Conductor 9922599225
Sign 11
Analytic cond. 345.424345.424
Root an. cond. 4.311104.31110
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 10·5-s − 14·7-s − 9·8-s + 10·10-s + 22·11-s − 22·13-s + 14·14-s − 47·16-s − 116·17-s + 102·19-s − 10·20-s − 22·22-s − 260·23-s + 75·25-s + 22·26-s − 14·28-s + 196·29-s + 150·31-s + 103·32-s + 116·34-s + 140·35-s − 96·37-s − 102·38-s + 90·40-s + 176·41-s + ⋯
L(s)  = 1  − 0.353·2-s + 1/8·4-s − 0.894·5-s − 0.755·7-s − 0.397·8-s + 0.316·10-s + 0.603·11-s − 0.469·13-s + 0.267·14-s − 0.734·16-s − 1.65·17-s + 1.23·19-s − 0.111·20-s − 0.213·22-s − 2.35·23-s + 3/5·25-s + 0.165·26-s − 0.0944·28-s + 1.25·29-s + 0.869·31-s + 0.568·32-s + 0.585·34-s + 0.676·35-s − 0.426·37-s − 0.435·38-s + 0.355·40-s + 0.670·41-s + ⋯

Functional equation

Λ(s)=(99225s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(99225s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9922599225    =    34⋅52⋅723^{4} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 345.424345.424
Root analytic conductor: 4.311104.31110
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 99225, ( :3/2,3/2), 1)(4,\ 99225,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3 1 1
5C1C_1 (1+pT)2 ( 1 + p T )^{2}
7C1C_1 (1+pT)2 ( 1 + p T )^{2}
good2D4D_{4} 1+T+p3T3+p6T4 1 + T + p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1−2pT+2718T2−2p4T3+p6T4 1 - 2 p T + 2718 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4}
13D4D_{4} 1+22T+4450T2+22p3T3+p6T4 1 + 22 T + 4450 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+116T+9030T2+116p3T3+p6T4 1 + 116 T + 9030 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−102T+13134T2−102p3T3+p6T4 1 - 102 T + 13134 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+260T+34734T2+260p3T3+p6T4 1 + 260 T + 34734 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−196T+20942T2−196p3T3+p6T4 1 - 196 T + 20942 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−150T+36542T2−150p3T3+p6T4 1 - 150 T + 36542 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+96T+82550T2+96p3T3+p6T4 1 + 96 T + 82550 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−176T−16914T2−176p3T3+p6T4 1 - 176 T - 16914 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+8pT+171958T2+8p4T3+p6T4 1 + 8 p T + 171958 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4}
47D4D_{4} 1+560T+248606T2+560p3T3+p6T4 1 + 560 T + 248606 T^{2} + 560 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+326T+204138T2+326p3T3+p6T4 1 + 326 T + 204138 T^{2} + 326 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−844T+474182T2−844p3T3+p6T4 1 - 844 T + 474182 T^{2} - 844 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+204T+455006T2+204p3T3+p6T4 1 + 204 T + 455006 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+104T+537670T2+104p3T3+p6T4 1 + 104 T + 537670 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+1670T+1384382T2+1670p3T3+p6T4 1 + 1670 T + 1384382 T^{2} + 1670 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+386T+152218T2+386p3T3+p6T4 1 + 386 T + 152218 T^{2} + 386 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+888T+1007454T2+888p3T3+p6T4 1 + 888 T + 1007454 T^{2} + 888 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+928T+600710T2+928p3T3+p6T4 1 + 928 T + 600710 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+588T+1495334T2+588p3T3+p6T4 1 + 588 T + 1495334 T^{2} + 588 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−522T+291282T2−522p3T3+p6T4 1 - 522 T + 291282 T^{2} - 522 p^{3} T^{3} + p^{6} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−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

−11.20852434211860158070550292526, −10.34301194838350969192057189641, −9.912917910576036660002778366986, −9.813598143111122291097187789564, −8.941796239597540146908615809167, −8.733023208634280286081840199193, −8.220907487845438659299340479149, −7.65912481459610089706425179296, −7.08460507343796884953169348500, −6.57599935206256681515534697106, −6.34244154936345227806499879321, −5.62836816283299934943447396523, −4.56804878502591952579584650807, −4.48637974831833031466241011187, −3.68410070749499476586243317733, −2.99071596766604524419226575158, −2.39909018157542419291794056314, −1.38628743103439624835248470911, 0, 0, 1.38628743103439624835248470911, 2.39909018157542419291794056314, 2.99071596766604524419226575158, 3.68410070749499476586243317733, 4.48637974831833031466241011187, 4.56804878502591952579584650807, 5.62836816283299934943447396523, 6.34244154936345227806499879321, 6.57599935206256681515534697106, 7.08460507343796884953169348500, 7.65912481459610089706425179296, 8.220907487845438659299340479149, 8.733023208634280286081840199193, 8.941796239597540146908615809167, 9.813598143111122291097187789564, 9.912917910576036660002778366986, 10.34301194838350969192057189641, 11.20852434211860158070550292526

Graph of the ZZ-function along the critical line