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Properties

Label 2-2385-265.179-c0-0-0
Degree 22
Conductor 23852385
Sign −0.915+0.402i-0.915 + 0.402i
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  + (0.239 + 1.30i)2-s + (−0.709 + 0.269i)4-s + (−0.787 + 0.616i)5-s + (0.165 + 0.273i)8-s + (−0.992 − 0.879i)10-s + (−0.885 + 0.784i)16-s + (−1.93 + 0.477i)17-s + (−0.542 + 0.244i)19-s + (0.392 − 0.649i)20-s + (1.25 + 1.25i)23-s + (0.239 − 0.970i)25-s + (−0.344 − 0.107i)31-s + (−0.983 − 0.770i)32-s + (−1.08 − 2.41i)34-s + (−0.448 − 0.649i)38-s + ⋯
L(s)  = 1  + (0.239 + 1.30i)2-s + (−0.709 + 0.269i)4-s + (−0.787 + 0.616i)5-s + (0.165 + 0.273i)8-s + (−0.992 − 0.879i)10-s + (−0.885 + 0.784i)16-s + (−1.93 + 0.477i)17-s + (−0.542 + 0.244i)19-s + (0.392 − 0.649i)20-s + (1.25 + 1.25i)23-s + (0.239 − 0.970i)25-s + (−0.344 − 0.107i)31-s + (−0.983 − 0.770i)32-s + (−1.08 − 2.41i)34-s + (−0.448 − 0.649i)38-s + ⋯

Functional equation

Λ(s)=(2385s/2 ΓC(s) L(s)=((−0.915+0.402i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2385s/2 ΓC(s) L(s)=((−0.915+0.402i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23852385    =    32⋅5⋅533^{2} \cdot 5 \cdot 53
Sign: −0.915+0.402i-0.915 + 0.402i
Analytic conductor: 1.190271.19027
Root analytic conductor: 1.090991.09099
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2385(1504,⋅)\chi_{2385} (1504, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2385, ( :0), −0.915+0.402i)(2,\ 2385,\ (\ :0),\ -0.915 + 0.402i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.88303317510.8830331751
L(12)L(\frac12) ≈\approx 0.88303317510.8830331751
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+(0.787−0.616i)T 1 + (0.787 - 0.616i)T
53 1+(0.517−0.855i)T 1 + (0.517 - 0.855i)T
good2 1+(−0.239−1.30i)T+(−0.935+0.354i)T2 1 + (-0.239 - 1.30i)T + (-0.935 + 0.354i)T^{2}
7 1+(−0.354−0.935i)T2 1 + (-0.354 - 0.935i)T^{2}
11 1+(−0.568−0.822i)T2 1 + (-0.568 - 0.822i)T^{2}
13 1+(0.748+0.663i)T2 1 + (0.748 + 0.663i)T^{2}
17 1+(1.93−0.477i)T+(0.885−0.464i)T2 1 + (1.93 - 0.477i)T + (0.885 - 0.464i)T^{2}
19 1+(0.542−0.244i)T+(0.663−0.748i)T2 1 + (0.542 - 0.244i)T + (0.663 - 0.748i)T^{2}
23 1+(−1.25−1.25i)T+iT2 1 + (-1.25 - 1.25i)T + iT^{2}
29 1+(−0.568+0.822i)T2 1 + (-0.568 + 0.822i)T^{2}
31 1+(0.344+0.107i)T+(0.822+0.568i)T2 1 + (0.344 + 0.107i)T + (0.822 + 0.568i)T^{2}
37 1+(0.120−0.992i)T2 1 + (0.120 - 0.992i)T^{2}
41 1+(−0.822+0.568i)T2 1 + (-0.822 + 0.568i)T^{2}
43 1+(0.120+0.992i)T2 1 + (0.120 + 0.992i)T^{2}
47 1+(0.814−0.0989i)T+(0.970−0.239i)T2 1 + (0.814 - 0.0989i)T + (0.970 - 0.239i)T^{2}
59 1+(0.970−0.239i)T2 1 + (0.970 - 0.239i)T^{2}
61 1+(0.885−0.535i)T+(0.464−0.885i)T2 1 + (0.885 - 0.535i)T + (0.464 - 0.885i)T^{2}
67 1+(−0.663−0.748i)T2 1 + (-0.663 - 0.748i)T^{2}
71 1+(0.992−0.120i)T2 1 + (0.992 - 0.120i)T^{2}
73 1+(−0.464−0.885i)T2 1 + (-0.464 - 0.885i)T^{2}
79 1+(−0.283+1.54i)T+(−0.935−0.354i)T2 1 + (-0.283 + 1.54i)T + (-0.935 - 0.354i)T^{2}
83 1+(−1.32+1.32i)T−iT2 1 + (-1.32 + 1.32i)T - iT^{2}
89 1+(0.885−0.464i)T2 1 + (0.885 - 0.464i)T^{2}
97 1+(0.970+0.239i)T2 1 + (0.970 + 0.239i)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

−9.132820056004798228695535349086, −8.678249012703092455269598689256, −7.72141401263324774080326120017, −7.30122054767848010849919870894, −6.50405632596202779261194004023, −6.01787847001079674687101653844, −4.84371600442663099110458877669, −4.28712406828641208492695449902, −3.22433025486469460291175337050, −1.99478031841275561431218751250, 0.52601406899038369302046951507, 1.92117541109124773072043756707, 2.81105361222663386250816415976, 3.78068482060397298208222838929, 4.57504960558875934854441287467, 5.00452051983255243214296090172, 6.60768046543073857662998970048, 7.05760322886142868587344496820, 8.236266740940774802530259467600, 8.895054642546187822685013353645

Graph of the ZZ-function along the critical line