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Properties

Label 2-390-39.8-c1-0-2
Degree 22
Conductor 390390
Sign −0.789−0.614i-0.789 - 0.614i
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 + 1.41i)3-s − 1.00i·4-s + (−0.707 + 0.707i)5-s + (−1.70 − 0.292i)6-s + (0.707 + 0.707i)8-s + (−1.00 + 2.82i)9-s − 1.00i·10-s + (1.41 + 1.41i)11-s + (1.41 − 1.00i)12-s + (−3 + 2i)13-s + (−1.70 − 0.292i)15-s − 1.00·16-s − 4.24·17-s + (−1.29 − 2.70i)18-s + (6 + 6i)19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.577 + 0.816i)3-s − 0.500i·4-s + (−0.316 + 0.316i)5-s + (−0.696 − 0.119i)6-s + (0.250 + 0.250i)8-s + (−0.333 + 0.942i)9-s − 0.316i·10-s + (0.426 + 0.426i)11-s + (0.408 − 0.288i)12-s + (−0.832 + 0.554i)13-s + (−0.440 − 0.0756i)15-s − 0.250·16-s − 1.02·17-s + (−0.304 − 0.638i)18-s + (1.37 + 1.37i)19-s + ⋯

Functional equation

Λ(s)=(390s/2 ΓC(s) L(s)=((−0.789−0.614i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(390s/2 ΓC(s+1/2) L(s)=((−0.789−0.614i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.789 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 390390    =    2⋅3⋅5⋅132 \cdot 3 \cdot 5 \cdot 13
Sign: −0.789−0.614i-0.789 - 0.614i
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), −0.789−0.614i)(2,\ 390,\ (\ :1/2),\ -0.789 - 0.614i)

Particular Values

L(1)L(1) ≈\approx 0.338504+0.986146i0.338504 + 0.986146i
L(12)L(\frac12) ≈\approx 0.338504+0.986146i0.338504 + 0.986146i
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+(−1−1.41i)T 1 + (-1 - 1.41i)T
5 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
13 1+(3−2i)T 1 + (3 - 2i)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+4.24T+17T2 1 + 4.24T + 17T^{2}
19 1+(−6−6i)T+19iT2 1 + (-6 - 6i)T + 19iT^{2}
23 1+1.41T+23T2 1 + 1.41T + 23T^{2}
29 1−1.41iT−29T2 1 - 1.41iT - 29T^{2}
31 1+(3+3i)T+31iT2 1 + (3 + 3i)T + 31iT^{2}
37 1+(−5+5i)T−37iT2 1 + (-5 + 5i)T - 37iT^{2}
41 1+(−7.07+7.07i)T−41iT2 1 + (-7.07 + 7.07i)T - 41iT^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1+(4.24+4.24i)T+47iT2 1 + (4.24 + 4.24i)T + 47iT^{2}
53 1−11.3iT−53T2 1 - 11.3iT - 53T^{2}
59 1+(2.82+2.82i)T+59iT2 1 + (2.82 + 2.82i)T + 59iT^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1+(−5−5i)T+67iT2 1 + (-5 - 5i)T + 67iT^{2}
71 1+(−5.65+5.65i)T−71iT2 1 + (-5.65 + 5.65i)T - 71iT^{2}
73 1+(−6+6i)T−73iT2 1 + (-6 + 6i)T - 73iT^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1+(−5.65+5.65i)T−83iT2 1 + (-5.65 + 5.65i)T - 83iT^{2}
89 1+(−4.24−4.24i)T+89iT2 1 + (-4.24 - 4.24i)T + 89iT^{2}
97 1+(−10−10i)T+97iT2 1 + (-10 - 10i)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

−11.40834894612124093803376400960, −10.55579055220919664340553580919, −9.554308198601143989746839679692, −9.177415289507952277780629733266, −7.88352141201085335872112352848, −7.31775538686263285416472727303, −5.99129514401020568825503883306, −4.74855835723566852369912650104, −3.75476446524374393398850826685, −2.19553677753245923300945769963, 0.76639050194924002143569331056, 2.39367046731321102593747668067, 3.46325832119581224401560127071, 4.93931939876387097826589277771, 6.52395162277621688711317129857, 7.42630613824105012773309796664, 8.232713470840360096108154024278, 9.111435551954309844953359546284, 9.776002181501443931890110735056, 11.25905466866640928362952689025

Graph of the ZZ-function along the critical line