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Properties

Label 2-390-13.10-c1-0-2
Degree $2$
Conductor $390$
Sign $0.964 - 0.265i$
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 + (−1.73 − i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)10-s + (5.59 − 3.23i)11-s + 0.999·12-s + (1 + 3.46i)13-s + 1.99·14-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 + (−0.654 − 0.377i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (1.68 − 0.974i)11-s + 0.288·12-s + (0.277 + 0.960i)13-s + 0.534·14-s + (0.223 − 0.129i)15-s + (−0.125 − 0.216i)16-s + (0.485 − 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15280 + 0.155531i\)
\(L(\frac12)\) \(\approx\) \(1.15280 + 0.155531i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 3.46i)T \)
good7 \( 1 + (1.73 + i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.59 + 3.23i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.46 - 3.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.86 + 3.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.133 + 0.232i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (-7.96 + 4.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.73 + i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.96 - 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.53iT - 47T^{2} \)
53 \( 1 - 0.928T + 53T^{2} \)
59 \( 1 + (-7.33 - 4.23i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.19 + 9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.92 - 5.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (10.7 + 6.19i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 + 8.92iT - 83T^{2} \)
89 \( 1 + (0.464 - 0.267i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.464 - 0.267i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.42391371656786079508589163953, −10.05719180685212250643458355281, −9.412425011472397068456333371022, −8.847346073073651922932644429919, −7.75357349541346647192320341784, −6.63623566718143719176609460135, −5.78617412331234509461318941352, −4.31498322252172755025692376795, −3.28280422320829006610625162403, −1.17735732094833667893933524257, 1.37082138207289870932682328220, 2.87138954784061684477933101437, 3.85457115966113509359783228067, 5.77407397996638689192526797988, 6.79082252273857290333450803593, 7.52100024155861497188637053801, 8.626997887928937813892154417959, 9.569670128619518044723870423904, 10.06633265487803078178147423637, 11.46402647924584353568484137663

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