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Properties

Label 2-2523-87.65-c0-0-1
Degree 22
Conductor 25232523
Sign 0.745−0.666i0.745 - 0.666i
Analytic cond. 1.259141.25914
Root an. cond. 1.122111.12211
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(2523s/2 ΓC(s) L(s)=((0.745−0.666i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2523s/2 ΓC(s) L(s)=((0.745−0.666i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25232523    =    3⋅2923 \cdot 29^{2}
Sign: 0.745−0.666i0.745 - 0.666i
Analytic conductor: 1.259141.25914
Root analytic conductor: 1.122111.12211
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2523(2327,⋅)\chi_{2523} (2327, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2523, ( :0), 0.745−0.666i)(2,\ 2523,\ (\ :0),\ 0.745 - 0.666i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.82462557120.8246255712
L(12)L(\frac12) ≈\approx 0.82462557120.8246255712
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)
bad3 1+(0.900+0.433i)T 1 + (0.900 + 0.433i)T
29 1 1
good2 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
5 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
7 1+(−1.45−0.702i)T+(0.623+0.781i)T2 1 + (-1.45 - 0.702i)T + (0.623 + 0.781i)T^{2}
11 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
13 1+(−0.385+0.483i)T+(−0.222−0.974i)T2 1 + (-0.385 + 0.483i)T + (-0.222 - 0.974i)T^{2}
17 1−T2 1 - T^{2}
19 1+(0.556−0.268i)T+(0.623−0.781i)T2 1 + (0.556 - 0.268i)T + (0.623 - 0.781i)T^{2}
23 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
31 1+(−0.360−1.57i)T+(−0.900+0.433i)T2 1 + (-0.360 - 1.57i)T + (-0.900 + 0.433i)T^{2}
37 1+(−0.385−0.483i)T+(−0.222+0.974i)T2 1 + (-0.385 - 0.483i)T + (-0.222 + 0.974i)T^{2}
41 1−T2 1 - T^{2}
43 1+(−0.360+1.57i)T+(−0.900−0.433i)T2 1 + (-0.360 + 1.57i)T + (-0.900 - 0.433i)T^{2}
47 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
53 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
59 1−T2 1 - T^{2}
61 1+(−1.45−0.702i)T+(0.623+0.781i)T2 1 + (-1.45 - 0.702i)T + (0.623 + 0.781i)T^{2}
67 1+(−0.385−0.483i)T+(−0.222+0.974i)T2 1 + (-0.385 - 0.483i)T + (-0.222 + 0.974i)T^{2}
71 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
73 1+(0.137−0.602i)T+(−0.900−0.433i)T2 1 + (0.137 - 0.602i)T + (-0.900 - 0.433i)T^{2}
79 1+(−0.385−0.483i)T+(−0.222+0.974i)T2 1 + (-0.385 - 0.483i)T + (-0.222 + 0.974i)T^{2}
83 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
89 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
97 1+(−1.45+0.702i)T+(0.623−0.781i)T2 1 + (-1.45 + 0.702i)T + (0.623 - 0.781i)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

−8.915196549844056530390481070249, −8.347702571146648837073108232421, −7.82260068812345610508584919213, −6.96140059069370864131346708423, −5.80787460890999824735638659309, −5.29292267914781341713093294937, −4.64603121360270580670117956783, −3.74086485718717801968062371829, −2.29316375908271317754043981375, −1.18746750062128877200237639197, 0.77893374106820349585612958900, 1.89589849606511577362516958430, 3.86483197461043431239572689782, 4.36837053153204769597183802131, 4.90503272174198410642076775863, 5.80571678180759842041311837681, 6.45637479248285023736467774111, 7.62391340941950229014517278692, 8.221923093350793731580860654309, 9.146326981478220710890500363103

Graph of the ZZ-function along the critical line