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Properties

Label 2-2523-87.35-c0-0-5
Degree 22
Conductor 25232523
Sign −0.510+0.859i-0.510 + 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+0.859i-0.510 + 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 1.9225821961.922582196
L(12)L(\frac12) ≈\approx 1.9225821961.922582196
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

−8.476916428660894949141321251000, −8.004983950815997236101789991679, −7.45062501746509790970548503566, −6.63921139502084078521776783551, −5.60146544357262729571112952604, −4.76503272919922634093086083256, −3.71849081307201234511565816427, −3.09780613640425390563840335533, −2.09428098455190613825654761652, −1.11613387356117617558569275510, 1.78415266742459309156314613085, 3.10479174910509789356280881722, 3.82716054622805759268503621344, 4.94915848960930964941216748287, 5.40751521242256998651233432627, 5.92236518718995601478334237267, 6.87988696960492054353713204678, 8.001730339250659605469497004542, 8.410449471510493092654716841845, 9.325040114977261236496363492783

Graph of the ZZ-function along the critical line