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Properties

Label 2-390-39.5-c1-0-7
Degree 22
Conductor 390390
Sign −0.315−0.948i-0.315 - 0.948i
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 + (−0.292 + 1.70i)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 + (−0.292 + 1.70i)15-s − 1.00·16-s + 4.24·17-s + (−2.70 + 1.29i)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 + (−0.119 + 0.696i)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 + (−0.0756 + 0.440i)15-s − 0.250·16-s + 1.02·17-s + (−0.638 + 0.304i)18-s + (1.37 − 1.37i)19-s + ⋯

Functional equation

Λ(s)=(390s/2 ΓC(s) L(s)=((−0.315−0.948i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(390s/2 ΓC(s+1/2) L(s)=((−0.315−0.948i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 390390    =    2⋅3⋅5⋅132 \cdot 3 \cdot 5 \cdot 13
Sign: −0.315−0.948i-0.315 - 0.948i
Analytic conductor: 3.114163.11416
Root analytic conductor: 1.764691.76469
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ390(161,⋅)\chi_{390} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 390, ( :1/2), −0.315−0.948i)(2,\ 390,\ (\ :1/2),\ -0.315 - 0.948i)

Particular Values

L(1)L(1) ≈\approx 1.22718+1.70213i1.22718 + 1.70213i
L(12)L(\frac12) ≈\approx 1.22718+1.70213i1.22718 + 1.70213i
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

−11.60157003759171037686269303124, −10.47166448253875639492295470800, −9.780081550135617731072317801203, −8.875247154207263964967485541464, −7.70516523525033162017941841644, −7.06258620780537083217495270098, −5.40099739122978266624222503302, −4.97863658782715270985198834109, −3.47676681822620453935412557895, −2.59503202606996395796708432105, 1.28938405490221187358980306374, 2.61819772181064849607171230985, 3.71625485650797373136142951594, 5.24985340103808300097915671590, 6.09605847366919804460148865623, 7.39520917610058093275361626968, 8.139806958442382058356299822849, 9.443284150768076408336505446444, 9.929238864907929281287238367994, 11.36629238556856990658273666865

Graph of the ZZ-function along the critical line