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Properties

Label 2-2523-87.35-c0-0-2
Degree 22
Conductor 25232523
Sign 0.510−0.859i0.510 - 0.859i
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  + (−0.623 + 0.781i)2-s + (−0.222 + 0.974i)3-s + (−0.623 − 0.781i)6-s + (0.222 − 0.974i)7-s + (−0.900 − 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.900 − 0.433i)11-s + (0.900 − 0.433i)13-s + (0.623 + 0.781i)14-s + (0.900 − 0.433i)16-s − 17-s + (0.900 − 0.433i)18-s + (0.900 + 0.433i)21-s + (−0.222 + 0.974i)22-s + (0.623 − 0.781i)24-s + (−0.222 − 0.974i)25-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)2-s + (−0.222 + 0.974i)3-s + (−0.623 − 0.781i)6-s + (0.222 − 0.974i)7-s + (−0.900 − 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.900 − 0.433i)11-s + (0.900 − 0.433i)13-s + (0.623 + 0.781i)14-s + (0.900 − 0.433i)16-s − 17-s + (0.900 − 0.433i)18-s + (0.900 + 0.433i)21-s + (−0.222 + 0.974i)22-s + (0.623 − 0.781i)24-s + (−0.222 − 0.974i)25-s + ⋯

Functional equation

Λ(s)=(2523s/2 ΓC(s) L(s)=((0.510−0.859i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2523s/2 ΓC(s) L(s)=((0.510−0.859i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25232523    =    3⋅2923 \cdot 29^{2}
Sign: 0.510−0.859i0.510 - 0.859i
Analytic conductor: 1.259141.25914
Root analytic conductor: 1.122111.12211
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2523(1949,⋅)\chi_{2523} (1949, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2523, ( :0), 0.510−0.859i)(2,\ 2523,\ (\ :0),\ 0.510 - 0.859i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.79391548090.7939154809
L(12)L(\frac12) ≈\approx 0.79391548090.7939154809
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+(0.222−0.974i)T 1 + (0.222 - 0.974i)T
29 1 1
good2 1+(0.623−0.781i)T+(−0.222−0.974i)T2 1 + (0.623 - 0.781i)T + (-0.222 - 0.974i)T^{2}
5 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
7 1+(−0.222+0.974i)T+(−0.900−0.433i)T2 1 + (-0.222 + 0.974i)T + (-0.900 - 0.433i)T^{2}
11 1+(−0.900+0.433i)T+(0.623−0.781i)T2 1 + (-0.900 + 0.433i)T + (0.623 - 0.781i)T^{2}
13 1+(−0.900+0.433i)T+(0.623−0.781i)T2 1 + (-0.900 + 0.433i)T + (0.623 - 0.781i)T^{2}
17 1+T+T2 1 + T + T^{2}
19 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
23 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
31 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
37 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
41 1−2T+T2 1 - 2T + T^{2}
43 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
47 1+(−0.900+0.433i)T+(0.623−0.781i)T2 1 + (-0.900 + 0.433i)T + (0.623 - 0.781i)T^{2}
53 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
59 1−T2 1 - T^{2}
61 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
67 1+(−0.900−0.433i)T+(0.623+0.781i)T2 1 + (-0.900 - 0.433i)T + (0.623 + 0.781i)T^{2}
71 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
73 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
79 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
83 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
89 1+(0.623−0.781i)T+(−0.222−0.974i)T2 1 + (0.623 - 0.781i)T + (-0.222 - 0.974i)T^{2}
97 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)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

−9.075290674130316140948668033697, −8.516866141326428388733053870213, −7.84795043138989476207676539733, −6.85977885846635879674009512125, −6.27691224766498971701174091929, −5.55316526519010959079418338599, −4.14087115939527983283314722695, −3.97010207629327529820515278278, −2.78662899477071849093818123149, −0.78766672838909503018676717996, 1.18838779969392990850082374514, 1.98173228874256899409299350788, 2.69511509960339429108581784128, 3.98933783498493783356251050583, 5.27532470345007444996105123508, 6.03733259571589403954651903229, 6.55554403976147136582449216020, 7.52370273865906531685091145504, 8.510537994458242502978476829168, 9.035057650702593410859713023437

Graph of the ZZ-function along the critical line