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Properties

Label 2-390-13.3-c1-0-4
Degree $2$
Conductor $390$
Sign $0.0128 - 0.999i$
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  + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + 5-s + (−0.499 + 0.866i)6-s + (1 − 1.73i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)10-s + (2.5 + 4.33i)11-s − 0.999·12-s + (−1 + 3.46i)13-s + 1.99·14-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (1 − 1.73i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + 0.447·5-s + (−0.204 + 0.353i)6-s + (0.377 − 0.654i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (0.753 + 1.30i)11-s − 0.288·12-s + (−0.277 + 0.960i)13-s + 0.534·14-s + (0.129 + 0.223i)15-s + (−0.125 − 0.216i)16-s + (0.242 − 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36005 + 1.34272i\)
\(L(\frac12)\) \(\approx\) \(1.36005 + 1.34272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 - T \)
13 \( 1 + (1 - 3.46i)T \)
good7 \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.5 + 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 11T + 31T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-7.5 + 12.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8 + 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.59851873382212028174123408380, −10.52579127363265289146473027770, −9.432965150138429998936672765786, −9.044523223018925566985447037164, −7.44217447050888354884752365617, −7.07782553058567051413845670594, −5.65834962314051177795458970594, −4.58281741629197366423045951803, −3.85803129405172033292355299156, −2.05629244924098450802776565712, 1.31667554819934426472056041060, 2.69869632305814641377915641814, 3.75465106684680250691141445322, 5.47788980204384098368486761399, 5.93254172882572168601355129205, 7.37403208304333244737649204443, 8.584713050959618961038837565638, 9.123968642820724329699847085597, 10.36376533340799082646462021513, 11.18442299744367113597707158431

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