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Properties

Label 2-390-39.8-c1-0-5
Degree 22
Conductor 390390
Sign 0.970−0.240i0.970 - 0.240i
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  + (−0.707 + 0.707i)2-s + (1.58 − 0.707i)3-s − 1.00i·4-s + (−0.707 + 0.707i)5-s + (−0.618 + 1.61i)6-s + (0.707 + 0.707i)8-s + (2.00 − 2.23i)9-s − 1.00i·10-s + (1.41 + 1.41i)11-s + (−0.707 − 1.58i)12-s + (3.58 − 0.418i)13-s + (−0.618 + 1.61i)15-s − 1.00·16-s + 1.64·17-s + (0.166 + 2.99i)18-s + (1.16 + 1.16i)19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.912 − 0.408i)3-s − 0.500i·4-s + (−0.316 + 0.316i)5-s + (−0.252 + 0.660i)6-s + (0.250 + 0.250i)8-s + (0.666 − 0.745i)9-s − 0.316i·10-s + (0.426 + 0.426i)11-s + (−0.204 − 0.456i)12-s + (0.993 − 0.116i)13-s + (−0.159 + 0.417i)15-s − 0.250·16-s + 0.398·17-s + (0.0393 + 0.706i)18-s + (0.266 + 0.266i)19-s + ⋯

Functional equation

Λ(s)=(390s/2 ΓC(s) L(s)=((0.970−0.240i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(390s/2 ΓC(s+1/2) L(s)=((0.970−0.240i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 390390    =    2⋅3⋅5⋅132 \cdot 3 \cdot 5 \cdot 13
Sign: 0.970−0.240i0.970 - 0.240i
Analytic conductor: 3.114163.11416
Root analytic conductor: 1.764691.76469
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ390(281,⋅)\chi_{390} (281, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 390, ( :1/2), 0.970−0.240i)(2,\ 390,\ (\ :1/2),\ 0.970 - 0.240i)

Particular Values

L(1)L(1) ≈\approx 1.45095+0.177139i1.45095 + 0.177139i
L(12)L(\frac12) ≈\approx 1.45095+0.177139i1.45095 + 0.177139i
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+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
3 1+(−1.58+0.707i)T 1 + (-1.58 + 0.707i)T
5 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
13 1+(−3.58+0.418i)T 1 + (-3.58 + 0.418i)T
good7 1−7iT2 1 - 7iT^{2}
11 1+(−1.41−1.41i)T+11iT2 1 + (-1.41 - 1.41i)T + 11iT^{2}
17 1−1.64T+17T2 1 - 1.64T + 17T^{2}
19 1+(−1.16−1.16i)T+19iT2 1 + (-1.16 - 1.16i)T + 19iT^{2}
23 1−4.47T+23T2 1 - 4.47T + 23T^{2}
29 1+4.47iT−29T2 1 + 4.47iT - 29T^{2}
31 1+(3+3i)T+31iT2 1 + (3 + 3i)T + 31iT^{2}
37 1+(4−4i)T−37iT2 1 + (4 - 4i)T - 37iT^{2}
41 1+(2.23−2.23i)T−41iT2 1 + (2.23 - 2.23i)T - 41iT^{2}
43 1+0.837iT−43T2 1 + 0.837iT - 43T^{2}
47 1+(−1.64−1.64i)T+47iT2 1 + (-1.64 - 1.64i)T + 47iT^{2}
53 1+1.41iT−53T2 1 + 1.41iT - 53T^{2}
59 1+(−3.05−3.05i)T+59iT2 1 + (-3.05 - 3.05i)T + 59iT^{2}
61 1+14.6T+61T2 1 + 14.6T + 61T^{2}
67 1+(−0.837−0.837i)T+67iT2 1 + (-0.837 - 0.837i)T + 67iT^{2}
71 1+(9.53−9.53i)T−71iT2 1 + (9.53 - 9.53i)T - 71iT^{2}
73 1+(−1.16+1.16i)T−73iT2 1 + (-1.16 + 1.16i)T - 73iT^{2}
79 1+10T+79T2 1 + 10T + 79T^{2}
83 1+(−5.65+5.65i)T−83iT2 1 + (-5.65 + 5.65i)T - 83iT^{2}
89 1+(5.06+5.06i)T+89iT2 1 + (5.06 + 5.06i)T + 89iT^{2}
97 1+(−5.16−5.16i)T+97iT2 1 + (-5.16 - 5.16i)T + 97iT^{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

−11.27082751892949899950542195139, −10.18933536108040880680020979912, −9.310069222986089719251893786688, −8.505313875858648815574922900895, −7.66465788939106789943577139889, −6.90558507922383242984379306906, −5.91149871057404279815297651021, −4.25924257738273254236956166580, −3.06273931917052652215031047567, −1.42048428454966644722053941760, 1.45708633879871916704372554995, 3.12390734432927115711334986736, 3.89312823559299478481531099645, 5.20444884915956434525320809054, 6.86913082653819243260356392906, 7.87472866141934935504261711543, 8.872509610274487890706877977613, 9.115290878989004299585195909691, 10.40913308891471711300387201442, 11.04161602440410691815654506459

Graph of the ZZ-function along the critical line