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Properties

Label 2-2600-104.51-c0-0-2
Degree $2$
Conductor $2600$
Sign $1$
Analytic cond. $1.29756$
Root an. cond. $1.13910$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1.29756\)
Root analytic conductor: \(1.13910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (51, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2600,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.230018859\)
\(L(\frac12)\) \(\approx\) \(1.230018859\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
13 \( 1 + T \)
good3 \( 1 + 1.73T + T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 - 1.73T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.73T + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 1.73T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line