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Properties

Label 2-740-740.203-c0-0-0
Degree 22
Conductor 740740
Sign 0.402+0.915i0.402 + 0.915i
Analytic cond. 0.3693080.369308
Root an. cond. 0.6077070.607707
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  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.939 − 0.342i)5-s + (0.500 − 0.866i)8-s + (−0.984 − 0.173i)9-s + (−0.939 − 0.342i)10-s + (−0.300 − 1.70i)13-s + (−0.939 + 0.342i)16-s + (1.85 + 0.326i)17-s + (0.642 + 0.766i)18-s + (0.499 + 0.866i)20-s + (0.766 − 0.642i)25-s + (−0.866 + 1.5i)26-s + (0.424 + 1.58i)29-s + (0.939 + 0.342i)32-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.939 − 0.342i)5-s + (0.500 − 0.866i)8-s + (−0.984 − 0.173i)9-s + (−0.939 − 0.342i)10-s + (−0.300 − 1.70i)13-s + (−0.939 + 0.342i)16-s + (1.85 + 0.326i)17-s + (0.642 + 0.766i)18-s + (0.499 + 0.866i)20-s + (0.766 − 0.642i)25-s + (−0.866 + 1.5i)26-s + (0.424 + 1.58i)29-s + (0.939 + 0.342i)32-s + ⋯

Functional equation

Λ(s)=(740s/2 ΓC(s) L(s)=((0.402+0.915i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(740s/2 ΓC(s) L(s)=((0.402+0.915i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 740740    =    22⋅5⋅372^{2} \cdot 5 \cdot 37
Sign: 0.402+0.915i0.402 + 0.915i
Analytic conductor: 0.3693080.369308
Root analytic conductor: 0.6077070.607707
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ740(203,⋅)\chi_{740} (203, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 740, ( :0), 0.402+0.915i)(2,\ 740,\ (\ :0),\ 0.402 + 0.915i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.74540495750.7454049575
L(12)L(\frac12) ≈\approx 0.74540495750.7454049575
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)
bad2 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
5 1+(−0.939+0.342i)T 1 + (-0.939 + 0.342i)T
37 1+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
good3 1+(0.984+0.173i)T2 1 + (0.984 + 0.173i)T^{2}
7 1+(−0.642+0.766i)T2 1 + (-0.642 + 0.766i)T^{2}
11 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
13 1+(0.300+1.70i)T+(−0.939+0.342i)T2 1 + (0.300 + 1.70i)T + (-0.939 + 0.342i)T^{2}
17 1+(−1.85−0.326i)T+(0.939+0.342i)T2 1 + (-1.85 - 0.326i)T + (0.939 + 0.342i)T^{2}
19 1+(−0.984−0.173i)T2 1 + (-0.984 - 0.173i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1+(−0.424−1.58i)T+(−0.866+0.5i)T2 1 + (-0.424 - 1.58i)T + (-0.866 + 0.5i)T^{2}
31 1−iT2 1 - iT^{2}
41 1+(0.673−0.118i)T+(0.939−0.342i)T2 1 + (0.673 - 0.118i)T + (0.939 - 0.342i)T^{2}
43 1−T2 1 - T^{2}
47 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
53 1+(0.469+0.218i)T+(0.642+0.766i)T2 1 + (0.469 + 0.218i)T + (0.642 + 0.766i)T^{2}
59 1+(−0.642−0.766i)T2 1 + (-0.642 - 0.766i)T^{2}
61 1+(0.657−0.939i)T+(−0.342−0.939i)T2 1 + (0.657 - 0.939i)T + (-0.342 - 0.939i)T^{2}
67 1+(0.642−0.766i)T2 1 + (0.642 - 0.766i)T^{2}
71 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
73 1+(1.36−1.36i)T−iT2 1 + (1.36 - 1.36i)T - iT^{2}
79 1+(0.642−0.766i)T2 1 + (0.642 - 0.766i)T^{2}
83 1+(−0.342+0.939i)T2 1 + (-0.342 + 0.939i)T^{2}
89 1+(0.766−1.64i)T+(−0.642−0.766i)T2 1 + (0.766 - 1.64i)T + (-0.642 - 0.766i)T^{2}
97 1+(1.70−0.984i)T+(0.5−0.866i)T2 1 + (1.70 - 0.984i)T + (0.5 - 0.866i)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

−10.29776744569413910367237493448, −9.746758031704562688262585902362, −8.725406834184771288192408357594, −8.179925549647441831291120289873, −7.17805794646791810437837496436, −5.83349196701273292691492765263, −5.22072874071745622412873736648, −3.43660816832380720117629417738, −2.69465447410502764290323074582, −1.16952663504902888006865938130, 1.68215166914304830762483112233, 2.87569636221968225739938126366, 4.71897853176888414933725195004, 5.72564433105895538204688455209, 6.33167879203969423404567351207, 7.27959131436953335663125500516, 8.184921411907399112112546221942, 9.142667816413862388465947885479, 9.721785065154202841368552316871, 10.42013829865048491058343455980

Graph of the ZZ-function along the critical line