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Properties

Label 2-2523-87.83-c0-0-3
Degree $2$
Conductor $2523$
Sign $0.505 + 0.862i$
Analytic cond. $1.25914$
Root an. cond. $1.12211$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.900 − 0.433i)3-s + (−0.900 − 0.433i)4-s + (1.45 − 0.702i)7-s + (0.623 − 0.781i)9-s − 12-s + (0.385 + 0.483i)13-s + (0.623 + 0.781i)16-s + (0.556 + 0.268i)19-s + (1.00 − 1.26i)21-s + (−0.900 − 0.433i)25-s + (0.222 − 0.974i)27-s − 1.61·28-s + (−0.360 + 1.57i)31-s + (−0.900 + 0.433i)36-s + (−0.385 + 0.483i)37-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)3-s + (−0.900 − 0.433i)4-s + (1.45 − 0.702i)7-s + (0.623 − 0.781i)9-s − 12-s + (0.385 + 0.483i)13-s + (0.623 + 0.781i)16-s + (0.556 + 0.268i)19-s + (1.00 − 1.26i)21-s + (−0.900 − 0.433i)25-s + (0.222 − 0.974i)27-s − 1.61·28-s + (−0.360 + 1.57i)31-s + (−0.900 + 0.433i)36-s + (−0.385 + 0.483i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $0.505 + 0.862i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (605, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ 0.505 + 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.622501367\)
\(L(\frac12)\) \(\approx\) \(1.622501367\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 \)
good2 \( 1 + (0.900 + 0.433i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T^{2} \)
7 \( 1 + (-1.45 + 0.702i)T + (0.623 - 0.781i)T^{2} \)
11 \( 1 + (0.222 - 0.974i)T^{2} \)
13 \( 1 + (-0.385 - 0.483i)T + (-0.222 + 0.974i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.556 - 0.268i)T + (0.623 + 0.781i)T^{2} \)
23 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (0.360 - 1.57i)T + (-0.900 - 0.433i)T^{2} \)
37 \( 1 + (0.385 - 0.483i)T + (-0.222 - 0.974i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (0.360 + 1.57i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.222 - 0.974i)T^{2} \)
53 \( 1 + (0.900 + 0.433i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (1.45 - 0.702i)T + (0.623 - 0.781i)T^{2} \)
67 \( 1 + (-0.385 + 0.483i)T + (-0.222 - 0.974i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.137 - 0.602i)T + (-0.900 + 0.433i)T^{2} \)
79 \( 1 + (0.385 - 0.483i)T + (-0.222 - 0.974i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.900 + 0.433i)T^{2} \)
97 \( 1 + (1.45 + 0.702i)T + (0.623 + 0.781i)T^{2} \)
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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

−8.662910380980273129675766685481, −8.429914716153723294370871298284, −7.57789303202997593532499470166, −6.90967590387871186151799681286, −5.76991897066098809406061444289, −4.86232964751820237449171831376, −4.16731774202116359722856969907, −3.42101858620527504179679209599, −1.87687120132592601505691267956, −1.21738931379680041686014310446, 1.58222723084269493121479587053, 2.67268250797174147637614521480, 3.63232903145438124951767923009, 4.43188672454354942351415512573, 5.10549972652372593353724698312, 5.81956522735286864871739641392, 7.41951431509092818453986098388, 8.000611891682724182290030324860, 8.320989204162595973518818971110, 9.282955628368280082501744523639

Graph of the $Z$-function along the critical line