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Properties

Label 2-2600-104.51-c0-0-2
Degree 22
Conductor 26002600
Sign 11
Analytic cond. 1.297561.29756
Root an. cond. 1.139101.13910
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.73·3-s + 4-s − 1.73·6-s − 7-s + 8-s + 1.99·9-s − 1.73·12-s − 13-s − 14-s + 16-s + 1.73·17-s + 1.99·18-s + 1.73·21-s − 1.73·24-s − 26-s − 1.73·27-s − 28-s + 32-s + 1.73·34-s + 1.99·36-s + 37-s + 1.73·39-s + 1.73·42-s + 1.73·43-s + 47-s − 1.73·48-s + ⋯
L(s)  = 1  + 2-s − 1.73·3-s + 4-s − 1.73·6-s − 7-s + 8-s + 1.99·9-s − 1.73·12-s − 13-s − 14-s + 16-s + 1.73·17-s + 1.99·18-s + 1.73·21-s − 1.73·24-s − 26-s − 1.73·27-s − 28-s + 32-s + 1.73·34-s + 1.99·36-s + 37-s + 1.73·39-s + 1.73·42-s + 1.73·43-s + 47-s − 1.73·48-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 26002600    =    23⋅52⋅132^{3} \cdot 5^{2} \cdot 13
Sign: 11
Analytic conductor: 1.297561.29756
Root analytic conductor: 1.139101.13910
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2600(51,⋅)\chi_{2600} (51, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2600, ( :0), 1)(2,\ 2600,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2300188591.230018859
L(12)L(\frac12) ≈\approx 1.2300188591.230018859
L(1)L(1) 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−T 1 - T
5 1 1
13 1+T 1 + T
good3 1+1.73T+T2 1 + 1.73T + T^{2}
7 1+T+T2 1 + T + T^{2}
11 1−T2 1 - T^{2}
17 1−1.73T+T2 1 - 1.73T + T^{2}
19 1−T2 1 - T^{2}
23 1−T2 1 - T^{2}
29 1−T2 1 - T^{2}
31 1+T2 1 + T^{2}
37 1−T+T2 1 - T + T^{2}
41 1−T2 1 - T^{2}
43 1−1.73T+T2 1 - 1.73T + T^{2}
47 1−T+T2 1 - T + T^{2}
53 1−T2 1 - T^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1−T2 1 - T^{2}
71 1−1.73T+T2 1 - 1.73T + T^{2}
73 1−T2 1 - T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1−T2 1 - T^{2}
97 1−T2 1 - T^{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

−9.522487958944607961864714630526, −7.79635052144518044016939335623, −7.22241604658048938930039206152, −6.47137201464964437154848055536, −5.82219834444028657957592824014, −5.34300238800893777115157010075, −4.51005877094029633809506795905, −3.63059863656341321432137262474, −2.55128594963832080186597886290, −0.994106912809867627051311840340, 0.994106912809867627051311840340, 2.55128594963832080186597886290, 3.63059863656341321432137262474, 4.51005877094029633809506795905, 5.34300238800893777115157010075, 5.82219834444028657957592824014, 6.47137201464964437154848055536, 7.22241604658048938930039206152, 7.79635052144518044016939335623, 9.522487958944607961864714630526

Graph of the ZZ-function along the critical line