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Properties

Label 2-2600-13.12-c1-0-62
Degree 22
Conductor 26002600
Sign −0.922+0.384i-0.922 + 0.384i
Analytic cond. 20.761120.7611
Root an. cond. 4.556434.55643
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.949·3-s − 3.85i·7-s − 2.09·9-s − 0.589i·11-s + (−1.38 − 3.32i)13-s + 3.66·17-s + 5.94i·19-s − 3.66i·21-s − 3.51·23-s − 4.83·27-s − 5.33·29-s − 8.71i·31-s − 0.559i·33-s − 1.85i·37-s + (−1.31 − 3.15i)39-s + ⋯
L(s)  = 1  + 0.547·3-s − 1.45i·7-s − 0.699·9-s − 0.177i·11-s + (−0.384 − 0.922i)13-s + 0.887·17-s + 1.36i·19-s − 0.798i·21-s − 0.733·23-s − 0.931·27-s − 0.991·29-s − 1.56i·31-s − 0.0974i·33-s − 0.305i·37-s + (−0.210 − 0.505i)39-s + ⋯

Functional equation

Λ(s)=(2600s/2 ΓC(s) L(s)=((−0.922+0.384i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2600s/2 ΓC(s+1/2) L(s)=((−0.922+0.384i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26002600    =    23⋅52⋅132^{3} \cdot 5^{2} \cdot 13
Sign: −0.922+0.384i-0.922 + 0.384i
Analytic conductor: 20.761120.7611
Root analytic conductor: 4.556434.55643
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2600(2001,⋅)\chi_{2600} (2001, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2600, ( :1/2), −0.922+0.384i)(2,\ 2600,\ (\ :1/2),\ -0.922 + 0.384i)

Particular Values

L(1)L(1) ≈\approx 0.96039295010.9603929501
L(12)L(\frac12) ≈\approx 0.96039295010.9603929501
L(32)L(\frac{3}{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}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
13 1+(1.38+3.32i)T 1 + (1.38 + 3.32i)T
good3 1−0.949T+3T2 1 - 0.949T + 3T^{2}
7 1+3.85iT−7T2 1 + 3.85iT - 7T^{2}
11 1+0.589iT−11T2 1 + 0.589iT - 11T^{2}
17 1−3.66T+17T2 1 - 3.66T + 17T^{2}
19 1−5.94iT−19T2 1 - 5.94iT - 19T^{2}
23 1+3.51T+23T2 1 + 3.51T + 23T^{2}
29 1+5.33T+29T2 1 + 5.33T + 29T^{2}
31 1+8.71iT−31T2 1 + 8.71iT - 31T^{2}
37 1+1.85iT−37T2 1 + 1.85iT - 37T^{2}
41 1−4.63iT−41T2 1 - 4.63iT - 41T^{2}
43 1−6.30T+43T2 1 - 6.30T + 43T^{2}
47 1−3.85iT−47T2 1 - 3.85iT - 47T^{2}
53 1+10.7T+53T2 1 + 10.7T + 53T^{2}
59 1+10.3iT−59T2 1 + 10.3iT - 59T^{2}
61 1+13.4T+61T2 1 + 13.4T + 61T^{2}
67 1−4.09iT−67T2 1 - 4.09iT - 67T^{2}
71 1−5.34iT−71T2 1 - 5.34iT - 71T^{2}
73 1+6.09iT−73T2 1 + 6.09iT - 73T^{2}
79 1+11.1T+79T2 1 + 11.1T + 79T^{2}
83 1−8.77iT−83T2 1 - 8.77iT - 83T^{2}
89 1+0.413iT−89T2 1 + 0.413iT - 89T^{2}
97 1+8.45iT−97T2 1 + 8.45iT - 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.198465932858962198663237580035, −7.87242420321189745966847755143, −7.36376985018316724279367243017, −6.06806236203967602069074025461, −5.61119758956842401658096704407, −4.33454757795301282537386655016, −3.63844329166586592136012323717, −2.90551638609907472071121191613, −1.60015709880850443246246656042, −0.27186865707895990728374793173, 1.78857750638616321217703597728, 2.61122089702861409736209211725, 3.30601151167861648601106398165, 4.53830469092636401313926339956, 5.39685056421901802658329815988, 6.01181537977732132483028712669, 6.98904893922557594997691441322, 7.79461102340777390680649290739, 8.662816841802284282220294918761, 9.108103923735082830078603907463

Graph of the ZZ-function along the critical line