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Properties

Label 2-2960-2960.2933-c0-0-0
Degree 22
Conductor 29602960
Sign 0.716−0.697i0.716 - 0.697i
Analytic cond. 1.477231.47723
Root an. cond. 1.215411.21541
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  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + 0.999·6-s + (−0.366 + 1.36i)7-s − 0.999·8-s + 0.999·10-s + (1 + i)11-s + (0.499 − 0.866i)12-s + (−0.5 − 0.866i)13-s + (0.999 + i)14-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.499 − 0.866i)20-s + (−1.36 + 0.366i)21-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + 0.999·6-s + (−0.366 + 1.36i)7-s − 0.999·8-s + 0.999·10-s + (1 + i)11-s + (0.499 − 0.866i)12-s + (−0.5 − 0.866i)13-s + (0.999 + i)14-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.499 − 0.866i)20-s + (−1.36 + 0.366i)21-s + ⋯

Functional equation

Λ(s)=(2960s/2 ΓC(s) L(s)=((0.716−0.697i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2960s/2 ΓC(s) L(s)=((0.716−0.697i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 29602960    =    24⋅5⋅372^{4} \cdot 5 \cdot 37
Sign: 0.716−0.697i0.716 - 0.697i
Analytic conductor: 1.477231.47723
Root analytic conductor: 1.215411.21541
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2960(2933,⋅)\chi_{2960} (2933, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2960, ( :0), 0.716−0.697i)(2,\ 2960,\ (\ :0),\ 0.716 - 0.697i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8295444871.829544487
L(12)L(\frac12) ≈\approx 1.8295444871.829544487
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
5 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
37 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good3 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
7 1+(0.366−1.36i)T+(−0.866−0.5i)T2 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2}
11 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
13 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
17 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
19 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
23 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
29 1+iT2 1 + iT^{2}
31 1+T+T2 1 + T + T^{2}
41 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
43 1+iT−T2 1 + iT - T^{2}
47 1−iT2 1 - iT^{2}
53 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
59 1+(−0.366−1.36i)T+(−0.866+0.5i)T2 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2}
61 1+(−1.36−0.366i)T+(0.866+0.5i)T2 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1+(−1.73+i)T+(0.5−0.866i)T2 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2}
73 1−iT2 1 - iT^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
89 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
97 1−iT2 1 - iT^{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

−9.268180963473405842042714995055, −8.823109352774628446698623875492, −7.46266844348945936290803356838, −6.51532197080721099875161462714, −5.71495938869540365615727762790, −5.15888680910782515941135938460, −3.94068508686296449729205165612, −3.48456386823877800941004806261, −2.51822585297652514895331950761, −1.91443953037833343238994347086, 0.933770522305192184577127610965, 2.14351800040779040775867802014, 3.46526693871285102050131988672, 4.23485356235168887897103097655, 4.89232779640381181704043729427, 6.10861246519775590578475009891, 6.58652281751298125637122947873, 7.20896097423484676339364969377, 8.061689875557413560755688129974, 8.505426219567040381271505592534

Graph of the ZZ-function along the critical line