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Properties

Label 4-10e2-1.1-c17e2-0-2
Degree 44
Conductor 100100
Sign 11
Analytic cond. 335.703335.703
Root an. cond. 4.280444.28044
Motivic weight 1717
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  − 512·2-s − 6.30e3·3-s + 1.96e5·4-s − 7.81e5·5-s + 3.22e6·6-s + 6.54e6·7-s − 6.71e7·8-s + 2.35e6·9-s + 4.00e8·10-s + 1.18e9·11-s − 1.24e9·12-s − 2.01e9·13-s − 3.35e9·14-s + 4.92e9·15-s + 2.14e10·16-s − 1.87e10·17-s − 1.20e9·18-s − 1.36e11·19-s − 1.53e11·20-s − 4.12e10·21-s − 6.08e11·22-s − 6.49e11·23-s + 4.23e11·24-s + 4.57e11·25-s + 1.03e12·26-s − 5.93e11·27-s + 1.28e12·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.555·3-s + 3/2·4-s − 0.894·5-s + 0.785·6-s + 0.429·7-s − 1.41·8-s + 0.0182·9-s + 1.26·10-s + 1.67·11-s − 0.832·12-s − 0.686·13-s − 0.606·14-s + 0.496·15-s + 5/4·16-s − 0.652·17-s − 0.0257·18-s − 1.84·19-s − 1.34·20-s − 0.238·21-s − 2.36·22-s − 1.72·23-s + 0.785·24-s + 3/5·25-s + 0.970·26-s − 0.404·27-s + 0.643·28-s + ⋯

Functional equation

Λ(s)=(100s/2 ΓC(s)2 L(s)=( Λ(18−s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}
Λ(s)=(100s/2 ΓC(s+17/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 100100    =    22⋅522^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 335.703335.703
Root analytic conductor: 4.280444.28044
Motivic weight: 1717
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 100, ( :17/2,17/2), 1)(4,\ 100,\ (\ :17/2, 17/2),\ 1)

Particular Values

L(9)L(9) == 00
L(12)L(\frac12) == 00
L(192)L(\frac{19}{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)
bad2C1C_1 (1+p8T)2 ( 1 + p^{8} T )^{2}
5C1C_1 (1+p8T)2 ( 1 + p^{8} T )^{2}
good3D4D_{4} 1+6308T+4159738p2T2+6308p17T3+p34T4 1 + 6308 T + 4159738 p^{2} T^{2} + 6308 p^{17} T^{3} + p^{34} T^{4}
7D4D_{4} 1−6543844T+6366439481202p2T2−6543844p17T3+p34T4 1 - 6543844 T + 6366439481202 p^{2} T^{2} - 6543844 p^{17} T^{3} + p^{34} T^{4}
11D4D_{4} 1−9829824p2T+10967934276982726p2T2−9829824p19T3+p34T4 1 - 9829824 p^{2} T + 10967934276982726 p^{2} T^{2} - 9829824 p^{19} T^{3} + p^{34} T^{4}
13D4D_{4} 1+155224556pT+87811455351436398p2T2+155224556p18T3+p34T4 1 + 155224556 p T + 87811455351436398 p^{2} T^{2} + 155224556 p^{18} T^{3} + p^{34} T^{4}
17D4D_{4} 1+1103272908pT+ 1 + 1103272908 p T + 87 ⁣⋯ ⁣7887\!\cdots\!78T2+1103272908p18T3+p34T4 T^{2} + 1103272908 p^{18} T^{3} + p^{34} T^{4}
19D4D_{4} 1+136704830600T+ 1 + 136704830600 T + 13 ⁣⋯ ⁣7813\!\cdots\!78T2+136704830600p17T3+p34T4 T^{2} + 136704830600 p^{17} T^{3} + p^{34} T^{4}
23D4D_{4} 1+649234170708T+ 1 + 649234170708 T + 27 ⁣⋯ ⁣2227\!\cdots\!22T2+649234170708p17T3+p34T4 T^{2} + 649234170708 p^{17} T^{3} + p^{34} T^{4}
29D4D_{4} 1+4696543420020T+ 1 + 4696543420020 T + 20 ⁣⋯ ⁣1820\!\cdots\!18T2+4696543420020p17T3+p34T4 T^{2} + 4696543420020 p^{17} T^{3} + p^{34} T^{4}
31D4D_{4} 1−7120867378744T+ 1 - 7120867378744 T + 30 ⁣⋯ ⁣0630\!\cdots\!06T2−7120867378744p17T3+p34T4 T^{2} - 7120867378744 p^{17} T^{3} + p^{34} T^{4}
37D4D_{4} 1−9933114637444T+ 1 - 9933114637444 T + 42 ⁣⋯ ⁣1842\!\cdots\!18T2−9933114637444p17T3+p34T4 T^{2} - 9933114637444 p^{17} T^{3} + p^{34} T^{4}
41D4D_{4} 1−9622711880844T+ 1 - 9622711880844 T + 40 ⁣⋯ ⁣4640\!\cdots\!46T2−9622711880844p17T3+p34T4 T^{2} - 9622711880844 p^{17} T^{3} + p^{34} T^{4}
43D4D_{4} 1−179265953764pT− 1 - 179265953764 p T - 61 ⁣⋯ ⁣3861\!\cdots\!38T2−179265953764p18T3+p34T4 T^{2} - 179265953764 p^{18} T^{3} + p^{34} T^{4}
47D4D_{4} 1+466072575837996T+ 1 + 466072575837996 T + 10 ⁣⋯ ⁣7810\!\cdots\!78T2+466072575837996p17T3+p34T4 T^{2} + 466072575837996 p^{17} T^{3} + p^{34} T^{4}
53D4D_{4} 1+623333120284428T+ 1 + 623333120284428 T + 33 ⁣⋯ ⁣2233\!\cdots\!22T2+623333120284428p17T3+p34T4 T^{2} + 623333120284428 p^{17} T^{3} + p^{34} T^{4}
59D4D_{4} 1−654616071995160T+ 1 - 654616071995160 T + 97 ⁣⋯ ⁣3897\!\cdots\!38T2−654616071995160p17T3+p34T4 T^{2} - 654616071995160 p^{17} T^{3} + p^{34} T^{4}
61D4D_{4} 1−1329040385162884T+ 1 - 1329040385162884 T + 16 ⁣⋯ ⁣0616\!\cdots\!06T2−1329040385162884p17T3+p34T4 T^{2} - 1329040385162884 p^{17} T^{3} + p^{34} T^{4}
67D4D_{4} 1+4249016411877596T+ 1 + 4249016411877596 T + 22 ⁣⋯ ⁣5822\!\cdots\!58T2+4249016411877596p17T3+p34T4 T^{2} + 4249016411877596 p^{17} T^{3} + p^{34} T^{4}
71D4D_{4} 1−2035131949347624T− 1 - 2035131949347624 T - 98 ⁣⋯ ⁣7498\!\cdots\!74T2−2035131949347624p17T3+p34T4 T^{2} - 2035131949347624 p^{17} T^{3} + p^{34} T^{4}
73D4D_{4} 1−2248593461347972T+ 1 - 2248593461347972 T + 96 ⁣⋯ ⁣0296\!\cdots\!02T2−2248593461347972p17T3+p34T4 T^{2} - 2248593461347972 p^{17} T^{3} + p^{34} T^{4}
79D4D_{4} 1−19188427678495120T+ 1 - 19188427678495120 T + 45 ⁣⋯ ⁣1845\!\cdots\!18T2−19188427678495120p17T3+p34T4 T^{2} - 19188427678495120 p^{17} T^{3} + p^{34} T^{4}
83D4D_{4} 1+18474649086611988T+ 1 + 18474649086611988 T + 91 ⁣⋯ ⁣8291\!\cdots\!82T2+18474649086611988p17T3+p34T4 T^{2} + 18474649086611988 p^{17} T^{3} + p^{34} T^{4}
89D4D_{4} 1+17458912916395020T+ 1 + 17458912916395020 T + 29 ⁣⋯ ⁣2229\!\cdots\!22pT2+17458912916395020p17T3+p34T4 p T^{2} + 17458912916395020 p^{17} T^{3} + p^{34} T^{4}
97D4D_{4} 1−195934325822490244T+ 1 - 195934325822490244 T + 21 ⁣⋯ ⁣5821\!\cdots\!58T2−195934325822490244p17T3+p34T4 T^{2} - 195934325822490244 p^{17} T^{3} + p^{34} 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

−16.44060109450586892843172728268, −15.80511956302180615323028788421, −14.75560750667871913677444486947, −14.63122790578675903841159792684, −12.97191703121270810045472509464, −12.03366484773966092165775879482, −11.49386042541845569500570278901, −11.09736117497202661028797722908, −10.00949088855829780957427183570, −9.293081749459928340961998145999, −8.330234956749635530572762170744, −7.83443484987907512307869010544, −6.65212381802034175078729787725, −6.24293214062582122949390919918, −4.58681730151072479810673840130, −3.73470910704275074307727234376, −2.16006779529385751767201227750, −1.41911298626837526464284591090, 0, 0, 1.41911298626837526464284591090, 2.16006779529385751767201227750, 3.73470910704275074307727234376, 4.58681730151072479810673840130, 6.24293214062582122949390919918, 6.65212381802034175078729787725, 7.83443484987907512307869010544, 8.330234956749635530572762170744, 9.293081749459928340961998145999, 10.00949088855829780957427183570, 11.09736117497202661028797722908, 11.49386042541845569500570278901, 12.03366484773966092165775879482, 12.97191703121270810045472509464, 14.63122790578675903841159792684, 14.75560750667871913677444486947, 15.80511956302180615323028788421, 16.44060109450586892843172728268

Graph of the ZZ-function along the critical line