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Properties

Label 2-2960-2960.2653-c0-0-0
Degree 22
Conductor 29602960
Sign −0.303+0.952i-0.303 + 0.952i
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  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + 0.999·6-s + (−1.36 + 0.366i)7-s − 0.999i·8-s − 0.999·10-s + (1 − i)11-s + (0.866 − 0.499i)12-s + (−0.866 − 0.5i)13-s + (−0.999 + i)14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.499i)20-s + (−1.36 − 0.366i)21-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + 0.999·6-s + (−1.36 + 0.366i)7-s − 0.999i·8-s − 0.999·10-s + (1 − i)11-s + (0.866 − 0.499i)12-s + (−0.866 − 0.5i)13-s + (−0.999 + i)14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.499i)20-s + (−1.36 − 0.366i)21-s + ⋯

Functional equation

Λ(s)=(2960s/2 ΓC(s) L(s)=((−0.303+0.952i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2960s/2 ΓC(s) L(s)=((−0.303+0.952i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 29602960    =    24⋅5⋅372^{4} \cdot 5 \cdot 37
Sign: −0.303+0.952i-0.303 + 0.952i
Analytic conductor: 1.477231.47723
Root analytic conductor: 1.215411.21541
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2960(2653,⋅)\chi_{2960} (2653, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2960, ( :0), −0.303+0.952i)(2,\ 2960,\ (\ :0),\ -0.303 + 0.952i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8589419241.858941924
L(12)L(\frac12) ≈\approx 1.8589419241.858941924
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+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
5 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
37 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
good3 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
7 1+(1.36−0.366i)T+(0.866−0.5i)T2 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2}
11 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
13 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)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+T+T2 1 + T + T^{2}
47 1−iT2 1 - iT^{2}
53 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)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

−8.909548938459224123826373486926, −8.150784001851725724846763338634, −6.98588224065946504294983450466, −6.37223620684964732547095212043, −5.48408188285151495736270382229, −4.52389485981336885695173449877, −3.71675972076837110817201387967, −3.22625421081321184733950654863, −2.59031578627343758143073763606, −0.75817892646865677465865449856, 2.00249803956130361674357637052, 2.93241339511422814208722120099, 3.58760827757220044749053971400, 4.23612689281309335451332676922, 5.25229012724900773029402960015, 6.51510787029417149746854569528, 6.95546480873060149046200675098, 7.40031510319039315161524298805, 8.021162907752934314809722603729, 9.193654007379919206475678910586

Graph of the ZZ-function along the critical line