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Properties

Label 2-390-13.4-c1-0-3
Degree $2$
Conductor $390$
Sign $0.354 - 0.934i$
Analytic cond. $3.11416$
Root an. cond. $1.76469$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.354 - 0.934i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.354 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62799 + 1.12348i\)
\(L(\frac12)\) \(\approx\) \(1.62799 + 1.12348i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 - iT \)
13 \( 1 + (3.35 + 1.32i)T \)
good7 \( 1 + (-4.02 + 2.32i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.81 - 2.20i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.96 - 4.02i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.488 - 0.845i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.15 + 3.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.44iT - 31T^{2} \)
37 \( 1 + (-3.28 - 1.89i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.31 + 3.64i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.358 - 0.620i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.75iT - 47T^{2} \)
53 \( 1 - 13.5T + 53T^{2} \)
59 \( 1 + (1.88 - 1.09i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.73 + 6.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.58 + 0.912i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.88 - 3.97i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 4.36iT - 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 + 3.51iT - 83T^{2} \)
89 \( 1 + (-7.07 - 4.08i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.9 + 6.91i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

Graph of the $Z$-function along the critical line