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Properties

Label 2-2523-87.38-c0-0-5
Degree 22
Conductor 25232523
Sign −0.235+0.971i-0.235 + 0.971i
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  + (0.781 − 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

Λ(s)=(2523s/2 ΓC(s) L(s)=((−0.235+0.971i) Λ‾(1−s)\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}
Λ(s)=(2523s/2 ΓC(s) L(s)=((−0.235+0.971i) Λ‾(1−s)\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: 22
Conductor: 25232523    =    3⋅2923 \cdot 29^{2}
Sign: −0.235+0.971i-0.235 + 0.971i
Analytic conductor: 1.259141.25914
Root analytic conductor: 1.122111.12211
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2523(1952,⋅)\chi_{2523} (1952, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2523, ( :0), −0.235+0.971i)(2,\ 2523,\ (\ :0),\ -0.235 + 0.971i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.97766441540.9776644154
L(12)L(\frac12) ≈\approx 0.97766441540.9776644154
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+(−0.781+0.623i)T 1 + (-0.781 + 0.623i)T
29 1 1
good2 1+(0.623−0.781i)T2 1 + (0.623 - 0.781i)T^{2}
5 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
7 1+(1.00+1.26i)T+(−0.222+0.974i)T2 1 + (1.00 + 1.26i)T + (-0.222 + 0.974i)T^{2}
11 1+(−0.900+0.433i)T2 1 + (-0.900 + 0.433i)T^{2}
13 1+(−0.137−0.602i)T+(−0.900+0.433i)T2 1 + (-0.137 - 0.602i)T + (-0.900 + 0.433i)T^{2}
17 1+T2 1 + T^{2}
19 1+(0.483+0.385i)T+(0.222+0.974i)T2 1 + (0.483 + 0.385i)T + (0.222 + 0.974i)T^{2}
23 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
31 1+(0.702+1.45i)T+(−0.623+0.781i)T2 1 + (0.702 + 1.45i)T + (-0.623 + 0.781i)T^{2}
37 1+(0.602+0.137i)T+(0.900+0.433i)T2 1 + (0.602 + 0.137i)T + (0.900 + 0.433i)T^{2}
41 1+T2 1 + T^{2}
43 1+(−0.702+1.45i)T+(−0.623−0.781i)T2 1 + (-0.702 + 1.45i)T + (-0.623 - 0.781i)T^{2}
47 1+(−0.900+0.433i)T2 1 + (-0.900 + 0.433i)T^{2}
53 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
59 1−T2 1 - T^{2}
61 1+(1.26−1.00i)T+(0.222−0.974i)T2 1 + (1.26 - 1.00i)T + (0.222 - 0.974i)T^{2}
67 1+(−0.137+0.602i)T+(−0.900−0.433i)T2 1 + (-0.137 + 0.602i)T + (-0.900 - 0.433i)T^{2}
71 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
73 1+(−0.268+0.556i)T+(−0.623−0.781i)T2 1 + (-0.268 + 0.556i)T + (-0.623 - 0.781i)T^{2}
79 1+(−0.602−0.137i)T+(0.900+0.433i)T2 1 + (-0.602 - 0.137i)T + (0.900 + 0.433i)T^{2}
83 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
89 1+(0.623−0.781i)T2 1 + (0.623 - 0.781i)T^{2}
97 1+(1.26+1.00i)T+(0.222+0.974i)T2 1 + (1.26 + 1.00i)T + (0.222 + 0.974i)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

−8.916351866415626470738327536223, −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 ZZ-function along the critical line