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Properties

Label 2-1183-13.12-c1-0-4
Degree 22
Conductor 11831183
Sign 0.554−0.832i0.554 - 0.832i
Analytic cond. 9.446309.44630
Root an. cond. 3.073483.07348
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  − 2.22i·2-s + 0.549·3-s − 2.92·4-s + 4.22i·5-s − 1.22i·6-s − i·7-s + 2.06i·8-s − 2.69·9-s + 9.36·10-s − 0.549i·11-s − 1.60·12-s − 2.22·14-s + 2.31i·15-s − 1.28·16-s + 2.37·17-s + 5.98i·18-s + ⋯
L(s)  = 1  − 1.56i·2-s + 0.317·3-s − 1.46·4-s + 1.88i·5-s − 0.498i·6-s − 0.377i·7-s + 0.728i·8-s − 0.899·9-s + 2.96·10-s − 0.165i·11-s − 0.464·12-s − 0.593·14-s + 0.598i·15-s − 0.320·16-s + 0.576·17-s + 1.41i·18-s + ⋯

Functional equation

Λ(s)=(1183s/2 ΓC(s) L(s)=((0.554−0.832i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1183s/2 ΓC(s+1/2) L(s)=((0.554−0.832i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11831183    =    7⋅1327 \cdot 13^{2}
Sign: 0.554−0.832i0.554 - 0.832i
Analytic conductor: 9.446309.44630
Root analytic conductor: 3.073483.07348
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1183(337,⋅)\chi_{1183} (337, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1183, ( :1/2), 0.554−0.832i)(2,\ 1183,\ (\ :1/2),\ 0.554 - 0.832i)

Particular Values

L(1)L(1) ≈\approx 0.68912636360.6891263636
L(12)L(\frac12) ≈\approx 0.68912636360.6891263636
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)
bad7 1+iT 1 + iT
13 1 1
good2 1+2.22iT−2T2 1 + 2.22iT - 2T^{2}
3 1−0.549T+3T2 1 - 0.549T + 3T^{2}
5 1−4.22iT−5T2 1 - 4.22iT - 5T^{2}
11 1+0.549iT−11T2 1 + 0.549iT - 11T^{2}
17 1−2.37T+17T2 1 - 2.37T + 17T^{2}
19 1−3.61iT−19T2 1 - 3.61iT - 19T^{2}
23 1+5.81T+23T2 1 + 5.81T + 23T^{2}
29 1+3.59T+29T2 1 + 3.59T + 29T^{2}
31 1−5.14iT−31T2 1 - 5.14iT - 31T^{2}
37 1+0.329iT−37T2 1 + 0.329iT - 37T^{2}
41 1−6.29iT−41T2 1 - 6.29iT - 41T^{2}
43 1+3.22T+43T2 1 + 3.22T + 43T^{2}
47 1−8.20iT−47T2 1 - 8.20iT - 47T^{2}
53 1−2.65T+53T2 1 - 2.65T + 53T^{2}
59 1+1.80iT−59T2 1 + 1.80iT - 59T^{2}
61 1−0.609T+61T2 1 - 0.609T + 61T^{2}
67 1+10.3iT−67T2 1 + 10.3iT - 67T^{2}
71 1−11.1iT−71T2 1 - 11.1iT - 71T^{2}
73 1−4.90iT−73T2 1 - 4.90iT - 73T^{2}
79 1+14.0T+79T2 1 + 14.0T + 79T^{2}
83 1−5.73iT−83T2 1 - 5.73iT - 83T^{2}
89 1+7.46iT−89T2 1 + 7.46iT - 89T^{2}
97 1−6.84iT−97T2 1 - 6.84iT - 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

−10.09561198270610859367309771999, −9.595339848450707508653893009574, −8.358932539305314319983662478075, −7.56865041467491066494608536648, −6.55063514696501701817312474353, −5.68684983047049584526945624676, −4.06603378116059055414201340466, −3.33189292157050453225091944178, −2.80031010405854466933858165534, −1.77481783129762205720600333027, 0.26772572594525316824304681138, 2.10858386739300633156347135544, 3.90088120398658949254749553004, 4.85571104418603014938920748418, 5.54898181080703554846974630671, 6.02403088277351423273605615847, 7.36855432376593242514391273212, 8.077393745146981867233621790219, 8.676925204716452761292030533410, 9.097445927867216656783678059541

Graph of the ZZ-function along the critical line