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Properties

Label 2-2523-3.2-c0-0-0
Degree 22
Conductor 25232523
Sign −i-i
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  + i·2-s − i·3-s + 6-s − 7-s + i·8-s − 9-s + i·11-s + 13-s − i·14-s − 16-s + i·17-s − i·18-s + i·21-s − 22-s + 24-s + 25-s + ⋯
L(s)  = 1  + i·2-s − i·3-s + 6-s − 7-s + i·8-s − 9-s + i·11-s + 13-s − i·14-s − 16-s + i·17-s − i·18-s + i·21-s − 22-s + 24-s + 25-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 25232523    =    3⋅2923 \cdot 29^{2}
Sign: −i-i
Analytic conductor: 1.259141.25914
Root analytic conductor: 1.122111.12211
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2523(842,⋅)\chi_{2523} (842, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2523, ( :0), −i)(2,\ 2523,\ (\ :0),\ -i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1267836451.126783645
L(12)L(\frac12) ≈\approx 1.1267836451.126783645
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+iT 1 + iT
29 1 1
good2 1−iT−T2 1 - iT - T^{2}
5 1−T2 1 - T^{2}
7 1+T+T2 1 + T + T^{2}
11 1−iT−T2 1 - iT - T^{2}
13 1−T+T2 1 - T + T^{2}
17 1−iT−T2 1 - iT - T^{2}
19 1+T2 1 + T^{2}
23 1−T2 1 - T^{2}
31 1+T2 1 + T^{2}
37 1+T2 1 + T^{2}
41 1−2iT−T2 1 - 2iT - T^{2}
43 1+T2 1 + T^{2}
47 1+iT−T2 1 + iT - T^{2}
53 1−T2 1 - T^{2}
59 1−T2 1 - T^{2}
61 1+T2 1 + T^{2}
67 1−T+T2 1 - T + T^{2}
71 1−T2 1 - T^{2}
73 1+T2 1 + T^{2}
79 1+T2 1 + T^{2}
83 1−T2 1 - T^{2}
89 1−iT−T2 1 - iT - T^{2}
97 1+T2 1 + 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

−8.914647264808119551147033118802, −8.302330534995812690361097935032, −7.63289350754837642783939638484, −6.75776201656374617482564800148, −6.48837348102456735980562929759, −5.81783706053121053892286880332, −4.85560986587378876305451612939, −3.54795345643235749065271537539, −2.55393028917965622868668693700, −1.49383854093713695742919085709, 0.76082110537880551539079789685, 2.45728836171047692851071837289, 3.31629230445045059492151244925, 3.61433403540286288046286177507, 4.72496452624142138876593834835, 5.80858522734923972651757789858, 6.40208766299634909919109050041, 7.32972770064118580662439226818, 8.646948265710912958429567992212, 9.046429866415798144627632653152

Graph of the ZZ-function along the critical line