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Properties

Label 2-2385-265.239-c0-0-1
Degree 22
Conductor 23852385
Sign 0.997−0.0759i0.997 - 0.0759i
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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0288 + 0.477i)2-s + (0.765 − 0.0929i)4-s + (0.297 − 0.954i)5-s + (0.152 + 0.833i)8-s + (0.464 + 0.114i)10-s + (0.354 − 0.0874i)16-s + (−0.587 − 0.851i)17-s + (1.12 + 1.43i)19-s + (0.138 − 0.758i)20-s + (0.501 − 0.501i)23-s + (−0.822 − 0.568i)25-s + (−0.0495 − 0.110i)31-s + (0.304 + 0.976i)32-s + (0.389 − 0.305i)34-s + (−0.653 + 0.578i)38-s + ⋯
L(s)  = 1  + (0.0288 + 0.477i)2-s + (0.765 − 0.0929i)4-s + (0.297 − 0.954i)5-s + (0.152 + 0.833i)8-s + (0.464 + 0.114i)10-s + (0.354 − 0.0874i)16-s + (−0.587 − 0.851i)17-s + (1.12 + 1.43i)19-s + (0.138 − 0.758i)20-s + (0.501 − 0.501i)23-s + (−0.822 − 0.568i)25-s + (−0.0495 − 0.110i)31-s + (0.304 + 0.976i)32-s + (0.389 − 0.305i)34-s + (−0.653 + 0.578i)38-s + ⋯

Functional equation

Λ(s)=(2385s/2 ΓC(s) L(s)=((0.997−0.0759i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0759i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2385s/2 ΓC(s) L(s)=((0.997−0.0759i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0759i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23852385    =    32⋅5⋅533^{2} \cdot 5 \cdot 53
Sign: 0.997−0.0759i0.997 - 0.0759i
Analytic conductor: 1.190271.19027
Root analytic conductor: 1.090991.09099
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2385(2359,⋅)\chi_{2385} (2359, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2385, ( :0), 0.997−0.0759i)(2,\ 2385,\ (\ :0),\ 0.997 - 0.0759i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6148704381.614870438
L(12)L(\frac12) ≈\approx 1.6148704381.614870438
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+(−0.297+0.954i)T 1 + (-0.297 + 0.954i)T
53 1+(−0.180+0.983i)T 1 + (-0.180 + 0.983i)T
good2 1+(−0.0288−0.477i)T+(−0.992+0.120i)T2 1 + (-0.0288 - 0.477i)T + (-0.992 + 0.120i)T^{2}
7 1+(0.120+0.992i)T2 1 + (0.120 + 0.992i)T^{2}
11 1+(0.748−0.663i)T2 1 + (0.748 - 0.663i)T^{2}
13 1+(0.970+0.239i)T2 1 + (0.970 + 0.239i)T^{2}
17 1+(0.587+0.851i)T+(−0.354+0.935i)T2 1 + (0.587 + 0.851i)T + (-0.354 + 0.935i)T^{2}
19 1+(−1.12−1.43i)T+(−0.239+0.970i)T2 1 + (-1.12 - 1.43i)T + (-0.239 + 0.970i)T^{2}
23 1+(−0.501+0.501i)T−iT2 1 + (-0.501 + 0.501i)T - iT^{2}
29 1+(0.748+0.663i)T2 1 + (0.748 + 0.663i)T^{2}
31 1+(0.0495+0.110i)T+(−0.663+0.748i)T2 1 + (0.0495 + 0.110i)T + (-0.663 + 0.748i)T^{2}
37 1+(0.885−0.464i)T2 1 + (0.885 - 0.464i)T^{2}
41 1+(0.663+0.748i)T2 1 + (0.663 + 0.748i)T^{2}
43 1+(0.885+0.464i)T2 1 + (0.885 + 0.464i)T^{2}
47 1+(0.731−1.39i)T+(−0.568−0.822i)T2 1 + (0.731 - 1.39i)T + (-0.568 - 0.822i)T^{2}
59 1+(−0.568−0.822i)T2 1 + (-0.568 - 0.822i)T^{2}
61 1+(−0.354+0.0649i)T+(0.935−0.354i)T2 1 + (-0.354 + 0.0649i)T + (0.935 - 0.354i)T^{2}
67 1+(0.239+0.970i)T2 1 + (0.239 + 0.970i)T^{2}
71 1+(−0.464+0.885i)T2 1 + (-0.464 + 0.885i)T^{2}
73 1+(−0.935−0.354i)T2 1 + (-0.935 - 0.354i)T^{2}
79 1+(−0.0359+0.593i)T+(−0.992−0.120i)T2 1 + (-0.0359 + 0.593i)T + (-0.992 - 0.120i)T^{2}
83 1+(1.40+1.40i)T+iT2 1 + (1.40 + 1.40i)T + iT^{2}
89 1+(−0.354+0.935i)T2 1 + (-0.354 + 0.935i)T^{2}
97 1+(−0.568+0.822i)T2 1 + (-0.568 + 0.822i)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

−9.095244422202913345855899370981, −8.252911723942391369079267859271, −7.66313588599315745951366431442, −6.82923730374100024852736173667, −6.02861485899934860772545707563, −5.33069723766634228845611754349, −4.66250490460406919758017236712, −3.40914457385602578794060368506, −2.29312658549316063587468870372, −1.27640884005111521869596845518, 1.46442545647634123029660167782, 2.53593080436798010126784061043, 3.13997899240565897296388195960, 4.06424811876752271780018862974, 5.30840366395604191106005674429, 6.17603124532711468109967746832, 6.97105880657911668277449235371, 7.29886434450623946883850810037, 8.364414297741997352984081898520, 9.403129043148983252772540334972

Graph of the ZZ-function along the critical line