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Properties

Label 2-2523-87.53-c0-0-0
Degree $2$
Conductor $2523$
Sign $-0.996 - 0.0833i$
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  + (−0.781 + 0.623i)2-s + (−0.974 + 0.222i)3-s + (0.623 − 0.781i)6-s + (0.222 + 0.974i)7-s + (−0.433 − 0.900i)8-s + (0.900 − 0.433i)9-s + (0.433 − 0.900i)11-s + (−0.900 − 0.433i)13-s + (−0.781 − 0.623i)14-s + (0.900 + 0.433i)16-s + i·17-s + (−0.433 + 0.900i)18-s + (−0.433 − 0.900i)21-s + (0.222 + 0.974i)22-s + (0.623 + 0.781i)24-s + (−0.222 + 0.974i)25-s + ⋯
L(s)  = 1  + (−0.781 + 0.623i)2-s + (−0.974 + 0.222i)3-s + (0.623 − 0.781i)6-s + (0.222 + 0.974i)7-s + (−0.433 − 0.900i)8-s + (0.900 − 0.433i)9-s + (0.433 − 0.900i)11-s + (−0.900 − 0.433i)13-s + (−0.781 − 0.623i)14-s + (0.900 + 0.433i)16-s + i·17-s + (−0.433 + 0.900i)18-s + (−0.433 − 0.900i)21-s + (0.222 + 0.974i)22-s + (0.623 + 0.781i)24-s + (−0.222 + 0.974i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $-0.996 - 0.0833i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (1619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ -0.996 - 0.0833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3609038172\)
\(L(\frac12)\) \(\approx\) \(0.3609038172\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.432424101667715798262135514773, −8.648128060663696962482393204613, −7.998157485043941215551613657760, −7.21655236272982117043129311631, −6.29254917107855502479411855945, −5.86857762944621285243571976464, −5.00126576754335610350657000264, −3.94749949634136722223036702698, −2.95772249378844592059610971721, −1.32782742937904939144006012620, 0.38772164171227691783980810970, 1.57415838426721702884835771536, 2.43089553112146849748320424500, 4.07219600912948579015888167892, 4.78638693613176157531277265664, 5.49825767891223805188778626607, 6.63935512492919216181765467299, 7.20112824119065944054086200593, 7.85874676179847066485943359252, 9.022933957641405759152257782877

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