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Properties

Label 2-2523-87.38-c0-0-5
Degree $2$
Conductor $2523$
Sign $-0.235 + 0.971i$
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)3-s + (−0.623 + 0.781i)4-s + (−1.00 − 1.26i)7-s + (0.222 − 0.974i)9-s + 0.999i·12-s + (0.137 + 0.602i)13-s + (−0.222 − 0.974i)16-s + (−0.483 − 0.385i)19-s + (−1.57 − 0.360i)21-s + (0.623 − 0.781i)25-s + (−0.433 − 0.900i)27-s + 1.61·28-s + (−0.702 − 1.45i)31-s + (0.623 + 0.781i)36-s + (−0.602 − 0.137i)37-s + ⋯
L(s)  = 1  + (0.781 − 0.623i)3-s + (−0.623 + 0.781i)4-s + (−1.00 − 1.26i)7-s + (0.222 − 0.974i)9-s + 0.999i·12-s + (0.137 + 0.602i)13-s + (−0.222 − 0.974i)16-s + (−0.483 − 0.385i)19-s + (−1.57 − 0.360i)21-s + (0.623 − 0.781i)25-s + (−0.433 − 0.900i)27-s + 1.61·28-s + (−0.702 − 1.45i)31-s + (0.623 + 0.781i)36-s + (−0.602 − 0.137i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $-0.235 + 0.971i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (1952, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ -0.235 + 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9776644154\)
\(L(\frac12)\) \(\approx\) \(0.9776644154\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.781 + 0.623i)T \)
29 \( 1 \)
good2 \( 1 + (0.623 - 0.781i)T^{2} \)
5 \( 1 + (-0.623 + 0.781i)T^{2} \)
7 \( 1 + (1.00 + 1.26i)T + (-0.222 + 0.974i)T^{2} \)
11 \( 1 + (-0.900 + 0.433i)T^{2} \)
13 \( 1 + (-0.137 - 0.602i)T + (-0.900 + 0.433i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.483 + 0.385i)T + (0.222 + 0.974i)T^{2} \)
23 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.702 + 1.45i)T + (-0.623 + 0.781i)T^{2} \)
37 \( 1 + (0.602 + 0.137i)T + (0.900 + 0.433i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-0.702 + 1.45i)T + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.900 + 0.433i)T^{2} \)
53 \( 1 + (-0.623 + 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (1.26 - 1.00i)T + (0.222 - 0.974i)T^{2} \)
67 \( 1 + (-0.137 + 0.602i)T + (-0.900 - 0.433i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.268 + 0.556i)T + (-0.623 - 0.781i)T^{2} \)
79 \( 1 + (-0.602 - 0.137i)T + (0.900 + 0.433i)T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.623 - 0.781i)T^{2} \)
97 \( 1 + (1.26 + 1.00i)T + (0.222 + 0.974i)T^{2} \)
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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

−8.916351866415626470738327536223, −8.077599723237157644386837386825, −7.30013179022852470924746788728, −6.91645854976125765015516903768, −6.03631453238743528186964693131, −4.51262246542365270254079404369, −3.91174556932971863591125252808, −3.26376568576919341952869216645, −2.23171053401064292144911824718, −0.58662283106179555050582212728, 1.70061124788177607234121540452, 2.87862136625238154875706835298, 3.52118520706816695812869374679, 4.64631538563051062024800012970, 5.39263506024133614144438476458, 6.00198671449767955669276453777, 6.95961641154329385418794679530, 8.171255167988821949463257887954, 8.756215618456770734374988086325, 9.303658381454771214791613818251

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