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Properties

Label 2-364-13.4-c1-0-0
Degree 22
Conductor 364364
Sign −0.968−0.247i-0.968 - 0.247i
Analytic cond. 2.906552.90655
Root an. cond. 1.704861.70486
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  + (−0.888 + 1.53i)3-s + 3.82i·5-s + (−0.866 + 0.5i)7-s + (−0.0783 − 0.135i)9-s + (−3.74 − 2.16i)11-s + (−0.930 − 3.48i)13-s + (−5.89 − 3.40i)15-s + (1.45 + 2.52i)17-s + (4.72 − 2.72i)19-s − 1.77i·21-s + (−0.307 + 0.531i)23-s − 9.65·25-s − 5.05·27-s + (−4.62 + 8.01i)29-s + 5.30i·31-s + ⋯
L(s)  = 1  + (−0.512 + 0.888i)3-s + 1.71i·5-s + (−0.327 + 0.188i)7-s + (−0.0261 − 0.0452i)9-s + (−1.12 − 0.652i)11-s + (−0.258 − 0.966i)13-s + (−1.52 − 0.878i)15-s + (0.353 + 0.612i)17-s + (1.08 − 0.626i)19-s − 0.387i·21-s + (−0.0640 + 0.110i)23-s − 1.93·25-s − 0.972·27-s + (−0.859 + 1.48i)29-s + 0.953i·31-s + ⋯

Functional equation

Λ(s)=(364s/2 ΓC(s) L(s)=((−0.968−0.247i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(364s/2 ΓC(s+1/2) L(s)=((−0.968−0.247i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 364364    =    22⋅7⋅132^{2} \cdot 7 \cdot 13
Sign: −0.968−0.247i-0.968 - 0.247i
Analytic conductor: 2.906552.90655
Root analytic conductor: 1.704861.70486
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ364(225,⋅)\chi_{364} (225, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 364, ( :1/2), −0.968−0.247i)(2,\ 364,\ (\ :1/2),\ -0.968 - 0.247i)

Particular Values

L(1)L(1) ≈\approx 0.100807+0.802975i0.100807 + 0.802975i
L(12)L(\frac12) ≈\approx 0.100807+0.802975i0.100807 + 0.802975i
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
7 1+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
13 1+(0.930+3.48i)T 1 + (0.930 + 3.48i)T
good3 1+(0.888−1.53i)T+(−1.5−2.59i)T2 1 + (0.888 - 1.53i)T + (-1.5 - 2.59i)T^{2}
5 1−3.82iT−5T2 1 - 3.82iT - 5T^{2}
11 1+(3.74+2.16i)T+(5.5+9.52i)T2 1 + (3.74 + 2.16i)T + (5.5 + 9.52i)T^{2}
17 1+(−1.45−2.52i)T+(−8.5+14.7i)T2 1 + (-1.45 - 2.52i)T + (-8.5 + 14.7i)T^{2}
19 1+(−4.72+2.72i)T+(9.5−16.4i)T2 1 + (-4.72 + 2.72i)T + (9.5 - 16.4i)T^{2}
23 1+(0.307−0.531i)T+(−11.5−19.9i)T2 1 + (0.307 - 0.531i)T + (-11.5 - 19.9i)T^{2}
29 1+(4.62−8.01i)T+(−14.5−25.1i)T2 1 + (4.62 - 8.01i)T + (-14.5 - 25.1i)T^{2}
31 1−5.30iT−31T2 1 - 5.30iT - 31T^{2}
37 1+(0.974+0.562i)T+(18.5+32.0i)T2 1 + (0.974 + 0.562i)T + (18.5 + 32.0i)T^{2}
41 1+(−10.4−6.01i)T+(20.5+35.5i)T2 1 + (-10.4 - 6.01i)T + (20.5 + 35.5i)T^{2}
43 1+(−0.641−1.11i)T+(−21.5+37.2i)T2 1 + (-0.641 - 1.11i)T + (-21.5 + 37.2i)T^{2}
47 1+5.68iT−47T2 1 + 5.68iT - 47T^{2}
53 1+3.96T+53T2 1 + 3.96T + 53T^{2}
59 1+(6.68−3.85i)T+(29.5−51.0i)T2 1 + (6.68 - 3.85i)T + (29.5 - 51.0i)T^{2}
61 1+(−7.17−12.4i)T+(−30.5+52.8i)T2 1 + (-7.17 - 12.4i)T + (-30.5 + 52.8i)T^{2}
67 1+(−0.468−0.270i)T+(33.5+58.0i)T2 1 + (-0.468 - 0.270i)T + (33.5 + 58.0i)T^{2}
71 1+(−7.96+4.60i)T+(35.5−61.4i)T2 1 + (-7.96 + 4.60i)T + (35.5 - 61.4i)T^{2}
73 1+7.36iT−73T2 1 + 7.36iT - 73T^{2}
79 1−0.331T+79T2 1 - 0.331T + 79T^{2}
83 1−16.6iT−83T2 1 - 16.6iT - 83T^{2}
89 1+(−2.55−1.47i)T+(44.5+77.0i)T2 1 + (-2.55 - 1.47i)T + (44.5 + 77.0i)T^{2}
97 1+(−8.36+4.82i)T+(48.5−84.0i)T2 1 + (-8.36 + 4.82i)T + (48.5 - 84.0i)T^{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

−11.40052290442522041131849763258, −10.67733549842854904604161840450, −10.41156167140864396637728205269, −9.443655186998453973670996462282, −7.914214564899512005451310169447, −7.13697622873416781205482161970, −5.86702885474462720782009423184, −5.19214722502258749934648609852, −3.50185041604711151370773670927, −2.78996180662873795073372542301, 0.57612086924706609804743883379, 1.98825808549093292114532682479, 4.13085900920106347679584761838, 5.18528876908070036755889214502, 6.03323074727835598177216618952, 7.42376276737744644668533207862, 7.897919884940108151936021964574, 9.394433986692652157610156202158, 9.712459435104048457494736237226, 11.38031282462232941162189149744

Graph of the ZZ-function along the critical line