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Properties

Label 2-115-5.4-c5-0-11
Degree 22
Conductor 115115
Sign 0.717−0.696i0.717 - 0.696i
Analytic cond. 18.444118.4441
Root an. cond. 4.294664.29466
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.77i·2-s + 16.5i·3-s − 28.3·4-s + (38.9 + 40.1i)5-s + 128.·6-s − 7.31i·7-s − 28.0i·8-s − 29.6·9-s + (311. − 302. i)10-s − 512.·11-s − 468. i·12-s + 603. i·13-s − 56.8·14-s + (−662. + 642. i)15-s − 1.12e3·16-s + 1.54e3i·17-s + ⋯
L(s)  = 1  − 1.37i·2-s + 1.05i·3-s − 0.887·4-s + (0.696 + 0.717i)5-s + 1.45·6-s − 0.0563i·7-s − 0.154i·8-s − 0.121·9-s + (0.986 − 0.956i)10-s − 1.27·11-s − 0.939i·12-s + 0.991i·13-s − 0.0774·14-s + (−0.760 + 0.737i)15-s − 1.10·16-s + 1.29i·17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((0.717−0.696i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+5/2) L(s)=((0.717−0.696i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: 0.717−0.696i0.717 - 0.696i
Analytic conductor: 18.444118.4441
Root analytic conductor: 4.294664.29466
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ115(24,⋅)\chi_{115} (24, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :5/2), 0.717−0.696i)(2,\ 115,\ (\ :5/2),\ 0.717 - 0.696i)

Particular Values

L(3)L(3) ≈\approx 1.7057411561.705741156
L(12)L(\frac12) ≈\approx 1.7057411561.705741156
L(72)L(\frac{7}{2}) 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+(−38.9−40.1i)T 1 + (-38.9 - 40.1i)T
23 1−529iT 1 - 529iT
good2 1+7.77iT−32T2 1 + 7.77iT - 32T^{2}
3 1−16.5iT−243T2 1 - 16.5iT - 243T^{2}
7 1+7.31iT−1.68e4T2 1 + 7.31iT - 1.68e4T^{2}
11 1+512.T+1.61e5T2 1 + 512.T + 1.61e5T^{2}
13 1−603.iT−3.71e5T2 1 - 603. iT - 3.71e5T^{2}
17 1−1.54e3iT−1.41e6T2 1 - 1.54e3iT - 1.41e6T^{2}
19 1−2.17e3T+2.47e6T2 1 - 2.17e3T + 2.47e6T^{2}
29 1+2.11e3T+2.05e7T2 1 + 2.11e3T + 2.05e7T^{2}
31 1+6.35e3T+2.86e7T2 1 + 6.35e3T + 2.86e7T^{2}
37 1+3.40e3iT−6.93e7T2 1 + 3.40e3iT - 6.93e7T^{2}
41 1−8.00e3T+1.15e8T2 1 - 8.00e3T + 1.15e8T^{2}
43 1−1.46e4iT−1.47e8T2 1 - 1.46e4iT - 1.47e8T^{2}
47 1−2.29e3iT−2.29e8T2 1 - 2.29e3iT - 2.29e8T^{2}
53 1−1.97e4iT−4.18e8T2 1 - 1.97e4iT - 4.18e8T^{2}
59 1−4.84e4T+7.14e8T2 1 - 4.84e4T + 7.14e8T^{2}
61 1+2.26e3T+8.44e8T2 1 + 2.26e3T + 8.44e8T^{2}
67 1+4.77e4iT−1.35e9T2 1 + 4.77e4iT - 1.35e9T^{2}
71 1+5.94e4T+1.80e9T2 1 + 5.94e4T + 1.80e9T^{2}
73 1+1.20e4iT−2.07e9T2 1 + 1.20e4iT - 2.07e9T^{2}
79 1+5.90e4T+3.07e9T2 1 + 5.90e4T + 3.07e9T^{2}
83 1−8.82e4iT−3.93e9T2 1 - 8.82e4iT - 3.93e9T^{2}
89 1−6.14e4T+5.58e9T2 1 - 6.14e4T + 5.58e9T^{2}
97 1+5.59e4iT−8.58e9T2 1 + 5.59e4iT - 8.58e9T^{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

−12.67440381456444354851817980420, −11.26104878594114844976988970464, −10.67175325187585483955577495022, −9.906480025673495044340984154370, −9.210602687510484403495837239746, −7.27777959910509069597051553058, −5.59865992238751739569236332933, −4.14693566911099187620510295807, −3.05102650454746914291374371272, −1.74097573759559664947671759342, 0.62126218084556640989897666572, 2.40238050959137791906522817291, 5.16077727678530791163660610043, 5.69561263286208808499789390957, 7.16842135174696814372738731983, 7.71267382719154389276273552270, 8.829000039741333777524552807061, 10.11604233167702972217796810931, 11.78073773620009542231417440208, 13.05099578560108901374199263475

Graph of the ZZ-function along the critical line