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Properties

Label 2-1160-145.12-c1-0-13
Degree 22
Conductor 11601160
Sign 0.348−0.937i0.348 - 0.937i
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  + 2.10·3-s + (−2.18 + 0.478i)5-s + (1.94 + 1.94i)7-s + 1.41·9-s + (0.883 + 0.883i)11-s + (2.69 + 2.69i)13-s + (−4.58 + 1.00i)15-s + 0.326i·17-s + (−5.28 + 5.28i)19-s + (4.09 + 4.09i)21-s + (−0.557 + 0.557i)23-s + (4.54 − 2.09i)25-s − 3.33·27-s + (−1.35 − 5.21i)29-s + (3.27 + 3.27i)31-s + ⋯
L(s)  = 1  + 1.21·3-s + (−0.976 + 0.214i)5-s + (0.736 + 0.736i)7-s + 0.471·9-s + (0.266 + 0.266i)11-s + (0.747 + 0.747i)13-s + (−1.18 + 0.259i)15-s + 0.0792i·17-s + (−1.21 + 1.21i)19-s + (0.893 + 0.893i)21-s + (−0.116 + 0.116i)23-s + (0.908 − 0.418i)25-s − 0.641·27-s + (−0.252 − 0.967i)29-s + (0.588 + 0.588i)31-s + ⋯

Functional equation

Λ(s)=(1160s/2 ΓC(s) L(s)=((0.348−0.937i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1160s/2 ΓC(s+1/2) L(s)=((0.348−0.937i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11601160    =    23⋅5⋅292^{3} \cdot 5 \cdot 29
Sign: 0.348−0.937i0.348 - 0.937i
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), 0.348−0.937i)(2,\ 1160,\ (\ :1/2),\ 0.348 - 0.937i)

Particular Values

L(1)L(1) ≈\approx 2.1106665082.110666508
L(12)L(\frac12) ≈\approx 2.1106665082.110666508
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+(2.18−0.478i)T 1 + (2.18 - 0.478i)T
29 1+(1.35+5.21i)T 1 + (1.35 + 5.21i)T
good3 1−2.10T+3T2 1 - 2.10T + 3T^{2}
7 1+(−1.94−1.94i)T+7iT2 1 + (-1.94 - 1.94i)T + 7iT^{2}
11 1+(−0.883−0.883i)T+11iT2 1 + (-0.883 - 0.883i)T + 11iT^{2}
13 1+(−2.69−2.69i)T+13iT2 1 + (-2.69 - 2.69i)T + 13iT^{2}
17 1−0.326iT−17T2 1 - 0.326iT - 17T^{2}
19 1+(5.28−5.28i)T−19iT2 1 + (5.28 - 5.28i)T - 19iT^{2}
23 1+(0.557−0.557i)T−23iT2 1 + (0.557 - 0.557i)T - 23iT^{2}
31 1+(−3.27−3.27i)T+31iT2 1 + (-3.27 - 3.27i)T + 31iT^{2}
37 1−6.69T+37T2 1 - 6.69T + 37T^{2}
41 1+(6.35−6.35i)T−41iT2 1 + (6.35 - 6.35i)T - 41iT^{2}
43 1+2.44T+43T2 1 + 2.44T + 43T^{2}
47 1−11.9T+47T2 1 - 11.9T + 47T^{2}
53 1+(−1.79+1.79i)T−53iT2 1 + (-1.79 + 1.79i)T - 53iT^{2}
59 1−6.73iT−59T2 1 - 6.73iT - 59T^{2}
61 1+(−5.50−5.50i)T+61iT2 1 + (-5.50 - 5.50i)T + 61iT^{2}
67 1+(−3.12+3.12i)T−67iT2 1 + (-3.12 + 3.12i)T - 67iT^{2}
71 1−3.62iT−71T2 1 - 3.62iT - 71T^{2}
73 1−11.1iT−73T2 1 - 11.1iT - 73T^{2}
79 1+(−5.17+5.17i)T−79iT2 1 + (-5.17 + 5.17i)T - 79iT^{2}
83 1+(−3.60+3.60i)T−83iT2 1 + (-3.60 + 3.60i)T - 83iT^{2}
89 1+(−10.8+10.8i)T−89iT2 1 + (-10.8 + 10.8i)T - 89iT^{2}
97 1+3.89T+97T2 1 + 3.89T + 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

−9.774811896197410877634230820611, −8.771400886931038899490712356891, −8.374529713219187162936773118439, −7.84446523544885539107334547150, −6.77582961032013623431286606667, −5.79002492895016118402652923669, −4.35440752662396868624274367060, −3.85516274642118343481263558149, −2.68894184871111146730505059729, −1.71866978040072504220136216780, 0.826707141795315107263985317847, 2.38132045155952489346617767103, 3.51872243692626643299599230570, 4.11818273547936216856812287205, 5.10256117341001122114583804376, 6.52131678105717021591850271786, 7.48340017791655134561201526292, 8.109448817202197421017328702972, 8.631381767314844192674889231725, 9.299463410468743271056410213659

Graph of the ZZ-function along the critical line