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Properties

Label 2-390-13.10-c1-0-6
Degree 22
Conductor 390390
Sign 0.354+0.934i0.354 + 0.934i
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  + (0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s − i·5-s + (−0.866 − 0.499i)6-s + (4.02 + 2.32i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (3.81 − 2.20i)11-s − 0.999·12-s + (−3.35 + 1.32i)13-s + 4.64·14-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (2 − 3.46i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s − 0.447i·5-s + (−0.353 − 0.204i)6-s + (1.52 + 0.877i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.158 − 0.273i)10-s + (1.14 − 0.663i)11-s − 0.288·12-s + (−0.930 + 0.366i)13-s + 1.24·14-s + (−0.223 + 0.129i)15-s + (−0.125 − 0.216i)16-s + (0.485 − 0.840i)17-s + ⋯

Functional equation

Λ(s)=(390s/2 ΓC(s) L(s)=((0.354+0.934i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(390s/2 ΓC(s+1/2) L(s)=((0.354+0.934i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.354 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 390390    =    2⋅3⋅5⋅132 \cdot 3 \cdot 5 \cdot 13
Sign: 0.354+0.934i0.354 + 0.934i
Analytic conductor: 3.114163.11416
Root analytic conductor: 1.764691.76469
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ390(361,⋅)\chi_{390} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 390, ( :1/2), 0.354+0.934i)(2,\ 390,\ (\ :1/2),\ 0.354 + 0.934i)

Particular Values

L(1)L(1) ≈\approx 1.62799−1.12348i1.62799 - 1.12348i
L(12)L(\frac12) ≈\approx 1.62799−1.12348i1.62799 - 1.12348i
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+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
5 1+iT 1 + iT
13 1+(3.35−1.32i)T 1 + (3.35 - 1.32i)T
good7 1+(−4.02−2.32i)T+(3.5+6.06i)T2 1 + (-4.02 - 2.32i)T + (3.5 + 6.06i)T^{2}
11 1+(−3.81+2.20i)T+(5.5−9.52i)T2 1 + (-3.81 + 2.20i)T + (5.5 - 9.52i)T^{2}
17 1+(−2+3.46i)T+(−8.5−14.7i)T2 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2}
19 1+(6.96+4.02i)T+(9.5+16.4i)T2 1 + (6.96 + 4.02i)T + (9.5 + 16.4i)T^{2}
23 1+(0.488+0.845i)T+(−11.5+19.9i)T2 1 + (0.488 + 0.845i)T + (-11.5 + 19.9i)T^{2}
29 1+(−2.15−3.73i)T+(−14.5+25.1i)T2 1 + (-2.15 - 3.73i)T + (-14.5 + 25.1i)T^{2}
31 1−6.44iT−31T2 1 - 6.44iT - 31T^{2}
37 1+(−3.28+1.89i)T+(18.5−32.0i)T2 1 + (-3.28 + 1.89i)T + (18.5 - 32.0i)T^{2}
41 1+(6.31−3.64i)T+(20.5−35.5i)T2 1 + (6.31 - 3.64i)T + (20.5 - 35.5i)T^{2}
43 1+(−0.358+0.620i)T+(−21.5−37.2i)T2 1 + (-0.358 + 0.620i)T + (-21.5 - 37.2i)T^{2}
47 1−9.75iT−47T2 1 - 9.75iT - 47T^{2}
53 1−13.5T+53T2 1 - 13.5T + 53T^{2}
59 1+(1.88+1.09i)T+(29.5+51.0i)T2 1 + (1.88 + 1.09i)T + (29.5 + 51.0i)T^{2}
61 1+(3.73−6.46i)T+(−30.5−52.8i)T2 1 + (3.73 - 6.46i)T + (-30.5 - 52.8i)T^{2}
67 1+(1.58−0.912i)T+(33.5−58.0i)T2 1 + (1.58 - 0.912i)T + (33.5 - 58.0i)T^{2}
71 1+(6.88+3.97i)T+(35.5+61.4i)T2 1 + (6.88 + 3.97i)T + (35.5 + 61.4i)T^{2}
73 1+4.36iT−73T2 1 + 4.36iT - 73T^{2}
79 1+14.9T+79T2 1 + 14.9T + 79T^{2}
83 1−3.51iT−83T2 1 - 3.51iT - 83T^{2}
89 1+(−7.07+4.08i)T+(44.5−77.0i)T2 1 + (-7.07 + 4.08i)T + (44.5 - 77.0i)T^{2}
97 1+(−11.9−6.91i)T+(48.5+84.0i)T2 1 + (-11.9 - 6.91i)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

−11.54665343075989333041231188851, −10.60056029780629329673421038406, −9.091157061229725669292686294597, −8.530392440369362893829674397850, −7.25293525731869865508733572228, −6.18557625203512561311730126408, −5.09800905159498476658462088282, −4.47061506942048281765745700127, −2.59524534362457127378917697732, −1.39277743682087190828840128906, 1.93973657953111178776505823991, 3.96084159815244289377413658045, 4.38847175936169729385174991285, 5.61153492403391391604303389884, 6.70496152593544578639994086916, 7.66216107986223151733029519108, 8.467368070231251108577606437796, 10.01995093726069598878330993587, 10.56156504402853270629040377454, 11.62919227687176000370831493050

Graph of the ZZ-function along the critical line