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Properties

Label 2-2523-87.80-c0-0-5
Degree 22
Conductor 25232523
Sign −0.0973+0.995i-0.0973 + 0.995i
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  + (0.433 − 0.900i)3-s + (0.900 − 0.433i)4-s + (−0.556 − 0.268i)7-s + (−0.623 − 0.781i)9-s − i·12-s + (1.00 − 1.26i)13-s + (0.623 − 0.781i)16-s + (0.702 + 1.45i)19-s + (−0.483 + 0.385i)21-s + (−0.900 + 0.433i)25-s + (−0.974 + 0.222i)27-s − 0.618·28-s + (0.602 − 0.137i)31-s + (−0.900 − 0.433i)36-s + (−1.26 + 1.00i)37-s + ⋯
L(s)  = 1  + (0.433 − 0.900i)3-s + (0.900 − 0.433i)4-s + (−0.556 − 0.268i)7-s + (−0.623 − 0.781i)9-s − i·12-s + (1.00 − 1.26i)13-s + (0.623 − 0.781i)16-s + (0.702 + 1.45i)19-s + (−0.483 + 0.385i)21-s + (−0.900 + 0.433i)25-s + (−0.974 + 0.222i)27-s − 0.618·28-s + (0.602 − 0.137i)31-s + (−0.900 − 0.433i)36-s + (−1.26 + 1.00i)37-s + ⋯

Functional equation

Λ(s)=(2523s/2 ΓC(s) L(s)=((−0.0973+0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0973 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2523s/2 ΓC(s) L(s)=((−0.0973+0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0973 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25232523    =    3⋅2923 \cdot 29^{2}
Sign: −0.0973+0.995i-0.0973 + 0.995i
Analytic conductor: 1.259141.25914
Root analytic conductor: 1.122111.12211
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2523(1037,⋅)\chi_{2523} (1037, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2523, ( :0), −0.0973+0.995i)(2,\ 2523,\ (\ :0),\ -0.0973 + 0.995i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6394492621.639449262
L(12)L(\frac12) ≈\approx 1.6394492621.639449262
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+(−0.433+0.900i)T 1 + (-0.433 + 0.900i)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+(0.556+0.268i)T+(0.623+0.781i)T2 1 + (0.556 + 0.268i)T + (0.623 + 0.781i)T^{2}
11 1+(−0.222−0.974i)T2 1 + (-0.222 - 0.974i)T^{2}
13 1+(−1.00+1.26i)T+(−0.222−0.974i)T2 1 + (-1.00 + 1.26i)T + (-0.222 - 0.974i)T^{2}
17 1+T2 1 + T^{2}
19 1+(−0.702−1.45i)T+(−0.623+0.781i)T2 1 + (-0.702 - 1.45i)T + (-0.623 + 0.781i)T^{2}
23 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
31 1+(−0.602+0.137i)T+(0.900−0.433i)T2 1 + (-0.602 + 0.137i)T + (0.900 - 0.433i)T^{2}
37 1+(1.26−1.00i)T+(0.222−0.974i)T2 1 + (1.26 - 1.00i)T + (0.222 - 0.974i)T^{2}
41 1+T2 1 + T^{2}
43 1+(0.602+0.137i)T+(0.900+0.433i)T2 1 + (0.602 + 0.137i)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+(−0.268+0.556i)T+(−0.623−0.781i)T2 1 + (-0.268 + 0.556i)T + (-0.623 - 0.781i)T^{2}
67 1+(−1.00−1.26i)T+(−0.222+0.974i)T2 1 + (-1.00 - 1.26i)T + (-0.222 + 0.974i)T^{2}
71 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
73 1+(1.57+0.360i)T+(0.900+0.433i)T2 1 + (1.57 + 0.360i)T + (0.900 + 0.433i)T^{2}
79 1+(−1.26+1.00i)T+(0.222−0.974i)T2 1 + (-1.26 + 1.00i)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+(−0.268−0.556i)T+(−0.623+0.781i)T2 1 + (-0.268 - 0.556i)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

−8.655777893140547970382949901468, −7.983727933174301285196993307849, −7.41146052023648670673670186475, −6.54866677544466390433302975550, −5.99480334113009229821182748474, −5.34998041720959555556160070022, −3.52877138425840324903883991580, −3.22231622867700364618763521247, −1.95655758618763198208599369894, −1.06639215672873844562521664396, 1.88415597651392593574899784920, 2.81365059398152046657690665603, 3.56593304083867216466966271891, 4.32871403383070492738463209082, 5.40144430059670496196630694877, 6.32692504091318866144920850982, 6.93227676679743047129208771899, 7.85511118235086202511699000384, 8.699469295496993048962784356550, 9.199121224682139446856165088137

Graph of the ZZ-function along the critical line