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Properties

Label 2-390-65.32-c1-0-2
Degree $2$
Conductor $390$
Sign $-0.966 - 0.256i$
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 + (−0.258 − 0.965i)3-s + (−0.499 − 0.866i)4-s + (1.08 + 1.95i)5-s + (0.965 + 0.258i)6-s + (−4.11 + 2.37i)7-s + 0.999·8-s + (−0.866 + 0.499i)9-s + (−2.23 − 0.0415i)10-s + (0.467 − 0.125i)11-s + (−0.707 + 0.707i)12-s + (−0.107 − 3.60i)13-s − 4.75i·14-s + (1.61 − 1.55i)15-s + (−0.5 + 0.866i)16-s + (−5.48 − 1.47i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.149 − 0.557i)3-s + (−0.249 − 0.433i)4-s + (0.483 + 0.875i)5-s + (0.394 + 0.105i)6-s + (−1.55 + 0.898i)7-s + 0.353·8-s + (−0.288 + 0.166i)9-s + (−0.706 − 0.0131i)10-s + (0.140 − 0.0377i)11-s + (−0.204 + 0.204i)12-s + (−0.0298 − 0.999i)13-s − 1.27i·14-s + (0.415 − 0.400i)15-s + (−0.125 + 0.216i)16-s + (−1.33 − 0.356i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.966 - 0.256i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0592954 + 0.455016i\)
\(L(\frac12)\) \(\approx\) \(0.0592954 + 0.455016i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-1.08 - 1.95i)T \)
13 \( 1 + (0.107 + 3.60i)T \)
good7 \( 1 + (4.11 - 2.37i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.467 + 0.125i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (5.48 + 1.47i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.99 - 7.44i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.08 - 1.09i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.30 - 1.33i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.58 - 3.58i)T - 31iT^{2} \)
37 \( 1 + (-4.49 - 2.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.85 + 6.92i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.97 - 11.0i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 3.72iT - 47T^{2} \)
53 \( 1 + (-2.48 + 2.48i)T - 53iT^{2} \)
59 \( 1 + (-7.30 - 1.95i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.66 + 2.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.630 - 1.09i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.7 - 2.86i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 4.98T + 73T^{2} \)
79 \( 1 + 10.0iT - 79T^{2} \)
83 \( 1 + 8.14iT - 83T^{2} \)
89 \( 1 + (-4.53 - 16.9i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-5.60 - 9.71i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.79479675394613517687740802549, −10.51760889210837415507828489670, −9.901527901591695492680401868182, −8.968007313186501811887100298461, −7.942411051514608220991539224459, −6.76124789718386659968138380396, −6.24803332630473254484768851132, −5.52836200292383125691824386846, −3.45666099356133919076220015962, −2.24093950144544418881722866632, 0.32222604996295037259214117806, 2.35215121437526180043727641450, 3.94352189719548910269725497991, 4.56528873513495015382735171704, 6.20159416943120474332617246160, 6.98448395320406460897975742874, 8.629401318693455359186108780126, 9.298695694762223166385020833605, 9.858623662552751200412898852584, 10.75955123328214470804327372097

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