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Properties

Label 2-2695-2695.2034-c0-0-1
Degree 22
Conductor 26952695
Sign 0.536−0.843i0.536 - 0.843i
Analytic cond. 1.344981.34498
Root an. cond. 1.159731.15973
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  + (1.07 − 0.332i)2-s + (0.222 − 0.151i)4-s + (0.988 + 0.149i)5-s + (−0.974 + 0.222i)7-s + (−0.513 + 0.643i)8-s + (−0.733 + 0.680i)9-s + (1.11 − 0.167i)10-s + (0.733 + 0.680i)11-s + (−0.302 + 1.32i)13-s + (−0.975 + 0.563i)14-s + (−0.437 + 1.11i)16-s + (0.116 − 1.55i)17-s + (−0.563 + 0.975i)18-s + (0.242 − 0.116i)20-s + (1.01 + 0.488i)22-s + ⋯
L(s)  = 1  + (1.07 − 0.332i)2-s + (0.222 − 0.151i)4-s + (0.988 + 0.149i)5-s + (−0.974 + 0.222i)7-s + (−0.513 + 0.643i)8-s + (−0.733 + 0.680i)9-s + (1.11 − 0.167i)10-s + (0.733 + 0.680i)11-s + (−0.302 + 1.32i)13-s + (−0.975 + 0.563i)14-s + (−0.437 + 1.11i)16-s + (0.116 − 1.55i)17-s + (−0.563 + 0.975i)18-s + (0.242 − 0.116i)20-s + (1.01 + 0.488i)22-s + ⋯

Functional equation

Λ(s)=(2695s/2 ΓC(s) L(s)=((0.536−0.843i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2695s/2 ΓC(s) L(s)=((0.536−0.843i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26952695    =    5⋅72⋅115 \cdot 7^{2} \cdot 11
Sign: 0.536−0.843i0.536 - 0.843i
Analytic conductor: 1.344981.34498
Root analytic conductor: 1.159731.15973
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2695(2034,⋅)\chi_{2695} (2034, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2695, ( :0), 0.536−0.843i)(2,\ 2695,\ (\ :0),\ 0.536 - 0.843i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8761259061.876125906
L(12)L(\frac12) ≈\approx 1.8761259061.876125906
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)
bad5 1+(−0.988−0.149i)T 1 + (-0.988 - 0.149i)T
7 1+(0.974−0.222i)T 1 + (0.974 - 0.222i)T
11 1+(−0.733−0.680i)T 1 + (-0.733 - 0.680i)T
good2 1+(−1.07+0.332i)T+(0.826−0.563i)T2 1 + (-1.07 + 0.332i)T + (0.826 - 0.563i)T^{2}
3 1+(0.733−0.680i)T2 1 + (0.733 - 0.680i)T^{2}
13 1+(0.302−1.32i)T+(−0.900−0.433i)T2 1 + (0.302 - 1.32i)T + (-0.900 - 0.433i)T^{2}
17 1+(−0.116+1.55i)T+(−0.988−0.149i)T2 1 + (-0.116 + 1.55i)T + (-0.988 - 0.149i)T^{2}
19 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(0.988−0.149i)T2 1 + (0.988 - 0.149i)T^{2}
29 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
31 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
37 1+(−0.365−0.930i)T2 1 + (-0.365 - 0.930i)T^{2}
41 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
43 1+(−0.367−0.460i)T+(−0.222+0.974i)T2 1 + (-0.367 - 0.460i)T + (-0.222 + 0.974i)T^{2}
47 1+(−0.826+0.563i)T2 1 + (-0.826 + 0.563i)T^{2}
53 1+(−0.365+0.930i)T2 1 + (-0.365 + 0.930i)T^{2}
59 1+(−1.44+0.218i)T+(0.955−0.294i)T2 1 + (-1.44 + 0.218i)T + (0.955 - 0.294i)T^{2}
61 1+(−0.365−0.930i)T2 1 + (-0.365 - 0.930i)T^{2}
67 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1+(1.48+0.716i)T+(0.623+0.781i)T2 1 + (1.48 + 0.716i)T + (0.623 + 0.781i)T^{2}
73 1+(−1.77−0.548i)T+(0.826+0.563i)T2 1 + (-1.77 - 0.548i)T + (0.826 + 0.563i)T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+(0.385+1.68i)T+(−0.900+0.433i)T2 1 + (0.385 + 1.68i)T + (-0.900 + 0.433i)T^{2}
89 1+(−1.40+1.29i)T+(0.0747−0.997i)T2 1 + (-1.40 + 1.29i)T + (0.0747 - 0.997i)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

−9.217751581804927843489273624937, −8.730321827998392522905369059341, −7.29633280743959052512943954319, −6.65382429122140804355185750449, −5.92176009082171256121099503409, −5.13019259890071532171057064473, −4.55342673004683964403700181148, −3.42987868208756827651263061956, −2.63527753926669738862162628560, −1.96329888386127398743874775533, 0.838596591906768165555204935712, 2.59143991588587596621714608358, 3.47944362583311279447212326701, 3.96727883701053603398769965666, 5.33195958692667344134438734336, 5.84882364156568518280994526160, 6.22886640986566064487790026573, 6.93908371099383626938663536189, 8.223121128574608197223042529396, 8.996653016515818986809688953897

Graph of the ZZ-function along the critical line