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Properties

Label 2-2385-265.154-c0-0-0
Degree 22
Conductor 23852385
Sign 0.129+0.991i0.129 + 0.991i
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  + (−1.78 + 0.556i)2-s + (2.05 − 1.41i)4-s + (0.0603 + 0.998i)5-s + (−1.72 + 2.20i)8-s + (−0.663 − 1.74i)10-s + (0.970 − 2.56i)16-s + (0.219 − 1.81i)17-s + (−1.68 + 0.308i)19-s + (1.53 + 1.96i)20-s + (−1.37 − 1.37i)23-s + (−0.992 + 0.120i)25-s + (−0.987 − 1.63i)31-s + (−0.140 + 2.31i)32-s + (0.614 + 3.35i)34-s + (2.83 − 1.48i)38-s + ⋯
L(s)  = 1  + (−1.78 + 0.556i)2-s + (2.05 − 1.41i)4-s + (0.0603 + 0.998i)5-s + (−1.72 + 2.20i)8-s + (−0.663 − 1.74i)10-s + (0.970 − 2.56i)16-s + (0.219 − 1.81i)17-s + (−1.68 + 0.308i)19-s + (1.53 + 1.96i)20-s + (−1.37 − 1.37i)23-s + (−0.992 + 0.120i)25-s + (−0.987 − 1.63i)31-s + (−0.140 + 2.31i)32-s + (0.614 + 3.35i)34-s + (2.83 − 1.48i)38-s + ⋯

Functional equation

Λ(s)=(2385s/2 ΓC(s) L(s)=((0.129+0.991i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2385s/2 ΓC(s) L(s)=((0.129+0.991i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23852385    =    32⋅5⋅533^{2} \cdot 5 \cdot 53
Sign: 0.129+0.991i0.129 + 0.991i
Analytic conductor: 1.190271.19027
Root analytic conductor: 1.090991.09099
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2385(154,⋅)\chi_{2385} (154, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2385, ( :0), 0.129+0.991i)(2,\ 2385,\ (\ :0),\ 0.129 + 0.991i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.20551627210.2055162721
L(12)L(\frac12) ≈\approx 0.20551627210.2055162721
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+(−0.0603−0.998i)T 1 + (-0.0603 - 0.998i)T
53 1+(0.616+0.787i)T 1 + (0.616 + 0.787i)T
good2 1+(1.78−0.556i)T+(0.822−0.568i)T2 1 + (1.78 - 0.556i)T + (0.822 - 0.568i)T^{2}
7 1+(0.568+0.822i)T2 1 + (0.568 + 0.822i)T^{2}
11 1+(−0.885+0.464i)T2 1 + (-0.885 + 0.464i)T^{2}
13 1+(0.354+0.935i)T2 1 + (0.354 + 0.935i)T^{2}
17 1+(−0.219+1.81i)T+(−0.970−0.239i)T2 1 + (-0.219 + 1.81i)T + (-0.970 - 0.239i)T^{2}
19 1+(1.68−0.308i)T+(0.935−0.354i)T2 1 + (1.68 - 0.308i)T + (0.935 - 0.354i)T^{2}
23 1+(1.37+1.37i)T+iT2 1 + (1.37 + 1.37i)T + iT^{2}
29 1+(−0.885−0.464i)T2 1 + (-0.885 - 0.464i)T^{2}
31 1+(0.987+1.63i)T+(−0.464+0.885i)T2 1 + (0.987 + 1.63i)T + (-0.464 + 0.885i)T^{2}
37 1+(−0.748−0.663i)T2 1 + (-0.748 - 0.663i)T^{2}
41 1+(0.464+0.885i)T2 1 + (0.464 + 0.885i)T^{2}
43 1+(−0.748+0.663i)T2 1 + (-0.748 + 0.663i)T^{2}
47 1+(−0.239−0.269i)T+(−0.120+0.992i)T2 1 + (-0.239 - 0.269i)T + (-0.120 + 0.992i)T^{2}
59 1+(−0.120+0.992i)T2 1 + (-0.120 + 0.992i)T^{2}
61 1+(−0.970−0.760i)T+(0.239+0.970i)T2 1 + (-0.970 - 0.760i)T + (0.239 + 0.970i)T^{2}
67 1+(−0.935−0.354i)T2 1 + (-0.935 - 0.354i)T^{2}
71 1+(0.663+0.748i)T2 1 + (0.663 + 0.748i)T^{2}
73 1+(−0.239+0.970i)T2 1 + (-0.239 + 0.970i)T^{2}
79 1+(−0.115−0.0359i)T+(0.822+0.568i)T2 1 + (-0.115 - 0.0359i)T + (0.822 + 0.568i)T^{2}
83 1+(1.16−1.16i)T−iT2 1 + (1.16 - 1.16i)T - iT^{2}
89 1+(−0.970−0.239i)T2 1 + (-0.970 - 0.239i)T^{2}
97 1+(−0.120−0.992i)T2 1 + (-0.120 - 0.992i)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

−8.948538778457495403352258606782, −8.151097723784066724647983399181, −7.60982921793906252795593484568, −6.78479906801186691114654375482, −6.35660492035544300983435072056, −5.50803839378407077116692890077, −4.07437749166031597566424699881, −2.59926199746871322793953336802, −2.04824447786473352903454634539, −0.23502453722026860380304678044, 1.50186613542306795558821365454, 1.96857504492954019451413168266, 3.42753698681277905194400625105, 4.27502985016607058597398762866, 5.68917092172578848183781194472, 6.43576954654920730912606816473, 7.43769744098338984922707761659, 8.250208583872593923696388521630, 8.524243151628716162490632797233, 9.273029015241460765326053100967

Graph of the ZZ-function along the critical line