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Properties

Label 2-390-65.4-c1-0-8
Degree 22
Conductor 390390
Sign 0.127+0.991i0.127 + 0.991i
Analytic cond. 3.114163.11416
Root an. cond. 1.764691.76469
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  + (−0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (1.26 + 1.84i)5-s + (0.866 − 0.499i)6-s + (−2.17 − 3.76i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (−2.22 + 0.178i)10-s + (−2.04 − 1.17i)11-s + 0.999i·12-s + (−3.18 − 1.69i)13-s + 4.34·14-s + (−0.178 − 2.22i)15-s + (−0.5 + 0.866i)16-s + (2.60 − 1.50i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.567 + 0.823i)5-s + (0.353 − 0.204i)6-s + (−0.821 − 1.42i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.704 + 0.0563i)10-s + (−0.615 − 0.355i)11-s + 0.288i·12-s + (−0.883 − 0.469i)13-s + 1.16·14-s + (−0.0459 − 0.575i)15-s + (−0.125 + 0.216i)16-s + (0.631 − 0.364i)17-s + ⋯

Functional equation

Λ(s)=(390s/2 ΓC(s) L(s)=((0.127+0.991i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(390s/2 ΓC(s+1/2) L(s)=((0.127+0.991i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 390390    =    2⋅3⋅5⋅132 \cdot 3 \cdot 5 \cdot 13
Sign: 0.127+0.991i0.127 + 0.991i
Analytic conductor: 3.114163.11416
Root analytic conductor: 1.764691.76469
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ390(199,⋅)\chi_{390} (199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 390, ( :1/2), 0.127+0.991i)(2,\ 390,\ (\ :1/2),\ 0.127 + 0.991i)

Particular Values

L(1)L(1) ≈\approx 0.439328−0.386460i0.439328 - 0.386460i
L(12)L(\frac12) ≈\approx 0.439328−0.386460i0.439328 - 0.386460i
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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
3 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
5 1+(−1.26−1.84i)T 1 + (-1.26 - 1.84i)T
13 1+(3.18+1.69i)T 1 + (3.18 + 1.69i)T
good7 1+(2.17+3.76i)T+(−3.5+6.06i)T2 1 + (2.17 + 3.76i)T + (-3.5 + 6.06i)T^{2}
11 1+(2.04+1.17i)T+(5.5+9.52i)T2 1 + (2.04 + 1.17i)T + (5.5 + 9.52i)T^{2}
17 1+(−2.60+1.50i)T+(8.5−14.7i)T2 1 + (-2.60 + 1.50i)T + (8.5 - 14.7i)T^{2}
19 1+(−0.585+0.338i)T+(9.5−16.4i)T2 1 + (-0.585 + 0.338i)T + (9.5 - 16.4i)T^{2}
23 1+(5.58+3.22i)T+(11.5+19.9i)T2 1 + (5.58 + 3.22i)T + (11.5 + 19.9i)T^{2}
29 1+(−4.82+8.35i)T+(−14.5−25.1i)T2 1 + (-4.82 + 8.35i)T + (-14.5 - 25.1i)T^{2}
31 1+7.11iT−31T2 1 + 7.11iT - 31T^{2}
37 1+(−3.74+6.48i)T+(−18.5−32.0i)T2 1 + (-3.74 + 6.48i)T + (-18.5 - 32.0i)T^{2}
41 1+(2.60+1.50i)T+(20.5+35.5i)T2 1 + (2.60 + 1.50i)T + (20.5 + 35.5i)T^{2}
43 1+(5.91−3.41i)T+(21.5−37.2i)T2 1 + (5.91 - 3.41i)T + (21.5 - 37.2i)T^{2}
47 1−5.61T+47T2 1 - 5.61T + 47T^{2}
53 1−9.43iT−53T2 1 - 9.43iT - 53T^{2}
59 1+(4.56−2.63i)T+(29.5−51.0i)T2 1 + (4.56 - 2.63i)T + (29.5 - 51.0i)T^{2}
61 1+(−2.15−3.73i)T+(−30.5+52.8i)T2 1 + (-2.15 - 3.73i)T + (-30.5 + 52.8i)T^{2}
67 1+(2.91−5.04i)T+(−33.5−58.0i)T2 1 + (2.91 - 5.04i)T + (-33.5 - 58.0i)T^{2}
71 1+(−2.52+1.45i)T+(35.5−61.4i)T2 1 + (-2.52 + 1.45i)T + (35.5 - 61.4i)T^{2}
73 1+7.67T+73T2 1 + 7.67T + 73T^{2}
79 1+3.74T+79T2 1 + 3.74T + 79T^{2}
83 1−10.3T+83T2 1 - 10.3T + 83T^{2}
89 1+(−4.15−2.39i)T+(44.5+77.0i)T2 1 + (-4.15 - 2.39i)T + (44.5 + 77.0i)T^{2}
97 1+(8.17+14.1i)T+(−48.5+84.0i)T2 1 + (8.17 + 14.1i)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

−10.71109617504674970876539528052, −10.15916638352330815746725625271, −9.640248955192947349365718590260, −7.83861970910049730257213569683, −7.38287668789310893529104093116, −6.38457298598746634983671582287, −5.70179462208914656363869880362, −4.23227505408380106987220355828, −2.66602949602715065844138439340, −0.44522657724787774130968135510, 1.84843441168044560255644499147, 3.15836002058783795694549406915, 4.84270354482072669378724094539, 5.51882273774921255350172985782, 6.65186088155334179099411705598, 8.179805733743700804553915571278, 9.044013287606965401288201406025, 9.819054857157749670881540932311, 10.28445355177561176020028081078, 11.81362523027541411414883973770

Graph of the ZZ-function along the critical line