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Properties

Label 2-2385-265.124-c0-0-0
Degree 22
Conductor 23852385
Sign 0.104−0.994i0.104 - 0.994i
Analytic cond. 1.190271.19027
Root an. cond. 1.090991.09099
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.40 − 0.851i)2-s + (0.794 + 1.51i)4-s + (−0.911 − 0.410i)5-s + (0.0704 − 1.16i)8-s + (0.935 + 1.35i)10-s + (−0.120 + 0.174i)16-s + (0.269 + 0.239i)17-s + (−1.17 + 0.366i)19-s + (−0.103 − 1.70i)20-s + (−0.170 + 0.170i)23-s + (0.663 + 0.748i)25-s + (−1.34 − 1.05i)31-s + (−0.746 + 0.335i)32-s + (−0.176 − 0.566i)34-s + (1.97 + 0.485i)38-s + ⋯
L(s)  = 1  + (−1.40 − 0.851i)2-s + (0.794 + 1.51i)4-s + (−0.911 − 0.410i)5-s + (0.0704 − 1.16i)8-s + (0.935 + 1.35i)10-s + (−0.120 + 0.174i)16-s + (0.269 + 0.239i)17-s + (−1.17 + 0.366i)19-s + (−0.103 − 1.70i)20-s + (−0.170 + 0.170i)23-s + (0.663 + 0.748i)25-s + (−1.34 − 1.05i)31-s + (−0.746 + 0.335i)32-s + (−0.176 − 0.566i)34-s + (1.97 + 0.485i)38-s + ⋯

Functional equation

Λ(s)=(2385s/2 ΓC(s) L(s)=((0.104−0.994i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2385s/2 ΓC(s) L(s)=((0.104−0.994i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23852385    =    32⋅5⋅533^{2} \cdot 5 \cdot 53
Sign: 0.104−0.994i0.104 - 0.994i
Analytic conductor: 1.190271.19027
Root analytic conductor: 1.090991.09099
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2385(919,⋅)\chi_{2385} (919, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2385, ( :0), 0.104−0.994i)(2,\ 2385,\ (\ :0),\ 0.104 - 0.994i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.12678484540.1267848454
L(12)L(\frac12) ≈\approx 0.12678484540.1267848454
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 1
5 1+(0.911+0.410i)T 1 + (0.911 + 0.410i)T
53 1+(−0.0603−0.998i)T 1 + (-0.0603 - 0.998i)T
good2 1+(1.40+0.851i)T+(0.464+0.885i)T2 1 + (1.40 + 0.851i)T + (0.464 + 0.885i)T^{2}
7 1+(0.885−0.464i)T2 1 + (0.885 - 0.464i)T^{2}
11 1+(0.970+0.239i)T2 1 + (0.970 + 0.239i)T^{2}
13 1+(−0.568−0.822i)T2 1 + (-0.568 - 0.822i)T^{2}
17 1+(−0.269−0.239i)T+(0.120+0.992i)T2 1 + (-0.269 - 0.239i)T + (0.120 + 0.992i)T^{2}
19 1+(1.17−0.366i)T+(0.822−0.568i)T2 1 + (1.17 - 0.366i)T + (0.822 - 0.568i)T^{2}
23 1+(0.170−0.170i)T−iT2 1 + (0.170 - 0.170i)T - iT^{2}
29 1+(0.970−0.239i)T2 1 + (0.970 - 0.239i)T^{2}
31 1+(1.34+1.05i)T+(0.239+0.970i)T2 1 + (1.34 + 1.05i)T + (0.239 + 0.970i)T^{2}
37 1+(−0.354−0.935i)T2 1 + (-0.354 - 0.935i)T^{2}
41 1+(−0.239+0.970i)T2 1 + (-0.239 + 0.970i)T^{2}
43 1+(−0.354+0.935i)T2 1 + (-0.354 + 0.935i)T^{2}
47 1+(0.556+0.210i)T+(0.748+0.663i)T2 1 + (0.556 + 0.210i)T + (0.748 + 0.663i)T^{2}
59 1+(0.748+0.663i)T2 1 + (0.748 + 0.663i)T^{2}
61 1+(0.120+0.00729i)T+(0.992+0.120i)T2 1 + (0.120 + 0.00729i)T + (0.992 + 0.120i)T^{2}
67 1+(−0.822−0.568i)T2 1 + (-0.822 - 0.568i)T^{2}
71 1+(−0.935−0.354i)T2 1 + (-0.935 - 0.354i)T^{2}
73 1+(−0.992+0.120i)T2 1 + (-0.992 + 0.120i)T^{2}
79 1+(1.56−0.943i)T+(0.464−0.885i)T2 1 + (1.56 - 0.943i)T + (0.464 - 0.885i)T^{2}
83 1+(−0.657−0.657i)T+iT2 1 + (-0.657 - 0.657i)T + iT^{2}
89 1+(0.120+0.992i)T2 1 + (0.120 + 0.992i)T^{2}
97 1+(0.748−0.663i)T2 1 + (0.748 - 0.663i)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

−9.230621518419971869305645252615, −8.722068405862252629030571538792, −7.922522555209376297619584411332, −7.58871280368738840867389836816, −6.51164636165695945018958817082, −5.37370138634795474347795209211, −4.20601554332691400308035640035, −3.48002552645656970309135117692, −2.34782142786850660196610659072, −1.30669633793806081000057514625, 0.14504643246485794346885214783, 1.74106156784258619109237606992, 3.12926103676406730124282761291, 4.17077287105807154577116802374, 5.25556879712805997574564764467, 6.32667615211836541978799209656, 6.90029979956585205179757411821, 7.50665601848430782879308879560, 8.275528283108012105617309778684, 8.721049391103953318165268937155

Graph of the ZZ-function along the critical line