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Properties

Label 2-2523-87.53-c0-0-0
Degree 22
Conductor 25232523
Sign −0.996−0.0833i-0.996 - 0.0833i
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)2-s + (−0.974 + 0.222i)3-s + (0.623 − 0.781i)6-s + (0.222 + 0.974i)7-s + (−0.433 − 0.900i)8-s + (0.900 − 0.433i)9-s + (0.433 − 0.900i)11-s + (−0.900 − 0.433i)13-s + (−0.781 − 0.623i)14-s + (0.900 + 0.433i)16-s + i·17-s + (−0.433 + 0.900i)18-s + (−0.433 − 0.900i)21-s + (0.222 + 0.974i)22-s + (0.623 + 0.781i)24-s + (−0.222 + 0.974i)25-s + ⋯
L(s)  = 1  + (−0.781 + 0.623i)2-s + (−0.974 + 0.222i)3-s + (0.623 − 0.781i)6-s + (0.222 + 0.974i)7-s + (−0.433 − 0.900i)8-s + (0.900 − 0.433i)9-s + (0.433 − 0.900i)11-s + (−0.900 − 0.433i)13-s + (−0.781 − 0.623i)14-s + (0.900 + 0.433i)16-s + i·17-s + (−0.433 + 0.900i)18-s + (−0.433 − 0.900i)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)=((−0.996−0.0833i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0833i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2523s/2 ΓC(s) L(s)=((−0.996−0.0833i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0833i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25232523    =    3⋅2923 \cdot 29^{2}
Sign: −0.996−0.0833i-0.996 - 0.0833i
Analytic conductor: 1.259141.25914
Root analytic conductor: 1.122111.12211
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2523(1619,⋅)\chi_{2523} (1619, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2523, ( :0), −0.996−0.0833i)(2,\ 2523,\ (\ :0),\ -0.996 - 0.0833i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.36090381720.3609038172
L(12)L(\frac12) ≈\approx 0.36090381720.3609038172
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+(0.974−0.222i)T 1 + (0.974 - 0.222i)T
29 1 1
good2 1+(0.781−0.623i)T+(0.222−0.974i)T2 1 + (0.781 - 0.623i)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+(−0.433+0.900i)T+(−0.623−0.781i)T2 1 + (-0.433 + 0.900i)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−iT−T2 1 - iT - 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 1−2iT−T2 1 - 2iT - T^{2}
43 1+(−0.222−0.974i)T2 1 + (-0.222 - 0.974i)T^{2}
47 1+(0.433−0.900i)T+(−0.623−0.781i)T2 1 + (0.433 - 0.900i)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+(0.781−0.623i)T+(0.222−0.974i)T2 1 + (0.781 - 0.623i)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

−9.432424101667715798262135514773, −8.648128060663696962482393204613, −7.998157485043941215551613657760, −7.21655236272982117043129311631, −6.29254917107855502479411855945, −5.86857762944621285243571976464, −5.00126576754335610350657000264, −3.94749949634136722223036702698, −2.95772249378844592059610971721, −1.32782742937904939144006012620, 0.38772164171227691783980810970, 1.57415838426721702884835771536, 2.43089553112146849748320424500, 4.07219600912948579015888167892, 4.78638693613176157531277265664, 5.49825767891223805188778626607, 6.63935512492919216181765467299, 7.20112824119065944054086200593, 7.85874676179847066485943359252, 9.022933957641405759152257782877

Graph of the ZZ-function along the critical line