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Properties

Label 2-1160-145.12-c1-0-0
Degree 22
Conductor 11601160
Sign −0.994+0.100i-0.994 + 0.100i
Analytic cond. 9.262649.26264
Root an. cond. 3.043453.04345
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.480·3-s + (−0.0382 + 2.23i)5-s + (0.312 + 0.312i)7-s − 2.76·9-s + (−1.55 − 1.55i)11-s + (−3.47 − 3.47i)13-s + (−0.0183 + 1.07i)15-s + 3.42i·17-s + (−2.53 + 2.53i)19-s + (0.150 + 0.150i)21-s + (−0.439 + 0.439i)23-s + (−4.99 − 0.171i)25-s − 2.77·27-s + (−4.36 − 3.15i)29-s + (1.56 + 1.56i)31-s + ⋯
L(s)  = 1  + 0.277·3-s + (−0.0171 + 0.999i)5-s + (0.118 + 0.118i)7-s − 0.923·9-s + (−0.468 − 0.468i)11-s + (−0.963 − 0.963i)13-s + (−0.00474 + 0.277i)15-s + 0.831i·17-s + (−0.582 + 0.582i)19-s + (0.0327 + 0.0327i)21-s + (−0.0916 + 0.0916i)23-s + (−0.999 − 0.0342i)25-s − 0.533·27-s + (−0.810 − 0.585i)29-s + (0.281 + 0.281i)31-s + ⋯

Functional equation

Λ(s)=(1160s/2 ΓC(s) L(s)=((−0.994+0.100i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1160s/2 ΓC(s+1/2) L(s)=((−0.994+0.100i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11601160    =    23⋅5⋅292^{3} \cdot 5 \cdot 29
Sign: −0.994+0.100i-0.994 + 0.100i
Analytic conductor: 9.262649.26264
Root analytic conductor: 3.043453.04345
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1160(737,⋅)\chi_{1160} (737, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1160, ( :1/2), −0.994+0.100i)(2,\ 1160,\ (\ :1/2),\ -0.994 + 0.100i)

Particular Values

L(1)L(1) ≈\approx 0.24525236980.2452523698
L(12)L(\frac12) ≈\approx 0.24525236980.2452523698
L(32)L(\frac{3}{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)
bad2 1 1
5 1+(0.0382−2.23i)T 1 + (0.0382 - 2.23i)T
29 1+(4.36+3.15i)T 1 + (4.36 + 3.15i)T
good3 1−0.480T+3T2 1 - 0.480T + 3T^{2}
7 1+(−0.312−0.312i)T+7iT2 1 + (-0.312 - 0.312i)T + 7iT^{2}
11 1+(1.55+1.55i)T+11iT2 1 + (1.55 + 1.55i)T + 11iT^{2}
13 1+(3.47+3.47i)T+13iT2 1 + (3.47 + 3.47i)T + 13iT^{2}
17 1−3.42iT−17T2 1 - 3.42iT - 17T^{2}
19 1+(2.53−2.53i)T−19iT2 1 + (2.53 - 2.53i)T - 19iT^{2}
23 1+(0.439−0.439i)T−23iT2 1 + (0.439 - 0.439i)T - 23iT^{2}
31 1+(−1.56−1.56i)T+31iT2 1 + (-1.56 - 1.56i)T + 31iT^{2}
37 1+0.0934T+37T2 1 + 0.0934T + 37T^{2}
41 1+(−0.878+0.878i)T−41iT2 1 + (-0.878 + 0.878i)T - 41iT^{2}
43 1−2.35T+43T2 1 - 2.35T + 43T^{2}
47 1+8.97T+47T2 1 + 8.97T + 47T^{2}
53 1+(−0.348+0.348i)T−53iT2 1 + (-0.348 + 0.348i)T - 53iT^{2}
59 1−0.108iT−59T2 1 - 0.108iT - 59T^{2}
61 1+(3.28+3.28i)T+61iT2 1 + (3.28 + 3.28i)T + 61iT^{2}
67 1+(−8.07+8.07i)T−67iT2 1 + (-8.07 + 8.07i)T - 67iT^{2}
71 1−14.1iT−71T2 1 - 14.1iT - 71T^{2}
73 1+2.92iT−73T2 1 + 2.92iT - 73T^{2}
79 1+(11.1−11.1i)T−79iT2 1 + (11.1 - 11.1i)T - 79iT^{2}
83 1+(8.68−8.68i)T−83iT2 1 + (8.68 - 8.68i)T - 83iT^{2}
89 1+(−0.0205+0.0205i)T−89iT2 1 + (-0.0205 + 0.0205i)T - 89iT^{2}
97 1+13.8T+97T2 1 + 13.8T + 97T^{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

−10.25964341867379457306460284112, −9.521501319394967914858586024216, −8.185493386187909791465954764950, −8.070551849682257635943472981047, −6.91299499580354607991925018163, −5.94418705252991036971815092695, −5.33821814364066734230740631153, −3.86384224887747473863712941974, −2.98498826914679225386099669503, −2.18027440707632015444451926492, 0.092351155728314496677547824895, 1.88206805644213940190859256053, 2.88796662882581659221741232047, 4.34779032443784190412923204666, 4.91860981042033568685913350304, 5.84481499273409002196918090215, 7.03895067520234310642606245840, 7.77732685695488859118222980406, 8.678760336117358142509338435690, 9.271827671102272801144697178737

Graph of the ZZ-function along the critical line