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Properties

Label 2-2600-13.12-c1-0-39
Degree 22
Conductor 26002600
Sign 0.566−0.824i0.566 - 0.824i
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  + 3.22·3-s + 1.65i·7-s + 7.41·9-s + 4.44i·11-s + (2.97 + 2.04i)13-s − 5.33·17-s + 0.472i·19-s + 5.33i·21-s − 1.08·23-s + 14.2·27-s − 5.50·29-s + 5.47i·31-s + 14.3i·33-s + 3.65i·37-s + (9.59 + 6.58i)39-s + ⋯
L(s)  = 1  + 1.86·3-s + 0.625i·7-s + 2.47·9-s + 1.34i·11-s + (0.824 + 0.566i)13-s − 1.29·17-s + 0.108i·19-s + 1.16i·21-s − 0.226·23-s + 2.73·27-s − 1.02·29-s + 0.982i·31-s + 2.49i·33-s + 0.600i·37-s + (1.53 + 1.05i)39-s + ⋯

Functional equation

Λ(s)=(2600s/2 ΓC(s) L(s)=((0.566−0.824i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2600s/2 ΓC(s+1/2) L(s)=((0.566−0.824i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26002600    =    23⋅52⋅132^{3} \cdot 5^{2} \cdot 13
Sign: 0.566−0.824i0.566 - 0.824i
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), 0.566−0.824i)(2,\ 2600,\ (\ :1/2),\ 0.566 - 0.824i)

Particular Values

L(1)L(1) ≈\approx 3.6479431493.647943149
L(12)L(\frac12) ≈\approx 3.6479431493.647943149
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+(−2.97−2.04i)T 1 + (-2.97 - 2.04i)T
good3 1−3.22T+3T2 1 - 3.22T + 3T^{2}
7 1−1.65iT−7T2 1 - 1.65iT - 7T^{2}
11 1−4.44iT−11T2 1 - 4.44iT - 11T^{2}
17 1+5.33T+17T2 1 + 5.33T + 17T^{2}
19 1−0.472iT−19T2 1 - 0.472iT - 19T^{2}
23 1+1.08T+23T2 1 + 1.08T + 23T^{2}
29 1+5.50T+29T2 1 + 5.50T + 29T^{2}
31 1−5.47iT−31T2 1 - 5.47iT - 31T^{2}
37 1−3.65iT−37T2 1 - 3.65iT - 37T^{2}
41 1+10.4iT−41T2 1 + 10.4iT - 41T^{2}
43 1−8.14T+43T2 1 - 8.14T + 43T^{2}
47 1+1.65iT−47T2 1 + 1.65iT - 47T^{2}
53 1+4.14T+53T2 1 + 4.14T + 53T^{2}
59 1−2.52iT−59T2 1 - 2.52iT - 59T^{2}
61 1−0.160T+61T2 1 - 0.160T + 61T^{2}
67 1−8.26iT−67T2 1 - 8.26iT - 67T^{2}
71 1+12.9iT−71T2 1 + 12.9iT - 71T^{2}
73 1+10.2iT−73T2 1 + 10.2iT - 73T^{2}
79 1+8.65T+79T2 1 + 8.65T + 79T^{2}
83 1+14.0iT−83T2 1 + 14.0iT - 83T^{2}
89 1−7.61iT−89T2 1 - 7.61iT - 89T^{2}
97 1+16.7iT−97T2 1 + 16.7iT - 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

−8.864630578476247571617954419195, −8.559050184675564210882326583730, −7.43022528014981289012115920204, −7.10168267019448967942293587728, −6.05334748711448665383507324276, −4.69919460389394493222247074389, −4.10597305803721693061867303141, −3.23004595915350682747225259684, −2.16109106444297900421409103862, −1.78893856720956338228675344294, 0.950782544221473328299216402275, 2.18974928210408963695823158547, 3.04671880008335309359555932698, 3.80431669689268855038315682810, 4.34970052427376633364869511863, 5.76504436768863972554372230652, 6.63926801697882489102602788264, 7.56139239434245827346451824401, 8.084219147984023337793633535352, 8.730062003508166046962727617055

Graph of the ZZ-function along the critical line