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Properties

Label 2-390-65.64-c1-0-1
Degree $2$
Conductor $390$
Sign $0.391 - 0.920i$
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  − 2-s − i·3-s + 4-s + (−2.17 + 0.539i)5-s + i·6-s − 0.630·7-s − 8-s − 9-s + (2.17 − 0.539i)10-s + 3.07i·11-s − i·12-s + (2.87 + 2.17i)13-s + 0.630·14-s + (0.539 + 2.17i)15-s + 16-s + 3.70i·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577i·3-s + 0.5·4-s + (−0.970 + 0.241i)5-s + 0.408i·6-s − 0.238·7-s − 0.353·8-s − 0.333·9-s + (0.686 − 0.170i)10-s + 0.928i·11-s − 0.288i·12-s + (0.798 + 0.601i)13-s + 0.168·14-s + (0.139 + 0.560i)15-s + 0.250·16-s + 0.899i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.391 - 0.920i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.535067 + 0.353812i\)
\(L(\frac12)\) \(\approx\) \(0.535067 + 0.353812i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + iT \)
5 \( 1 + (2.17 - 0.539i)T \)
13 \( 1 + (-2.87 - 2.17i)T \)
good7 \( 1 + 0.630T + 7T^{2} \)
11 \( 1 - 3.07iT - 11T^{2} \)
17 \( 1 - 3.70iT - 17T^{2} \)
19 \( 1 - 0.290iT - 19T^{2} \)
23 \( 1 - 1.70iT - 23T^{2} \)
29 \( 1 - 0.447T + 29T^{2} \)
31 \( 1 - 6.68iT - 31T^{2} \)
37 \( 1 - 5.07T + 37T^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 - 5.75iT - 43T^{2} \)
47 \( 1 + 2.73T + 47T^{2} \)
53 \( 1 - 14.0iT - 53T^{2} \)
59 \( 1 + 9.02iT - 59T^{2} \)
61 \( 1 + 3.26T + 61T^{2} \)
67 \( 1 + 7.75T + 67T^{2} \)
71 \( 1 + 1.65iT - 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + 3.23T + 83T^{2} \)
89 \( 1 + 12.3iT - 89T^{2} \)
97 \( 1 + 8.20T + 97T^{2} \)
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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

−11.39066421169809790118300598362, −10.68903485152492507474785695882, −9.580622700475346066511898112344, −8.564490984054016487467588894698, −7.80964979834511407520257027043, −6.94964091104125596297336542112, −6.16256376006583740012697109310, −4.43858772875803094844806083539, −3.16175894264660280978586602361, −1.54114778183158116704486577621, 0.56287584350859621638684160636, 2.95874401367370896424782214228, 3.95279575150939540275416391622, 5.33066338832679045462462377031, 6.47860038903327482563555876290, 7.72473893680566216771478497891, 8.440832421676011869792809173111, 9.214454406603735495467680192372, 10.23512825022561796458774710725, 11.19273169763895204180295123915

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