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Properties

Label 2-390-65.37-c1-0-4
Degree $2$
Conductor $390$
Sign $0.438 + 0.898i$
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 + (−0.965 − 0.258i)3-s + (0.499 − 0.866i)4-s + (−0.00342 − 2.23i)5-s + (0.965 − 0.258i)6-s + (−1.88 + 3.25i)7-s + 0.999i·8-s + (0.866 + 0.499i)9-s + (1.12 + 1.93i)10-s + (4.81 + 1.28i)11-s + (−0.707 + 0.707i)12-s + (−0.440 − 3.57i)13-s − 3.76i·14-s + (−0.575 + 2.16i)15-s + (−0.5 − 0.866i)16-s + (−1.49 − 5.59i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.557 − 0.149i)3-s + (0.249 − 0.433i)4-s + (−0.00152 − 0.999i)5-s + (0.394 − 0.105i)6-s + (−0.710 + 1.23i)7-s + 0.353i·8-s + (0.288 + 0.166i)9-s + (0.354 + 0.611i)10-s + (1.45 + 0.388i)11-s + (−0.204 + 0.204i)12-s + (−0.122 − 0.992i)13-s − 1.00i·14-s + (−0.148 + 0.557i)15-s + (−0.125 − 0.216i)16-s + (−0.363 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.438 + 0.898i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.438 + 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.603178 - 0.376683i\)
\(L(\frac12)\) \(\approx\) \(0.603178 - 0.376683i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (0.00342 + 2.23i)T \)
13 \( 1 + (0.440 + 3.57i)T \)
good7 \( 1 + (1.88 - 3.25i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.81 - 1.28i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.49 + 5.59i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.910 + 3.39i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.49 + 5.57i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.67 + 2.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.51 + 2.51i)T + 31iT^{2} \)
37 \( 1 + (-0.851 - 1.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.10 + 7.85i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.601 + 0.161i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 9.18T + 47T^{2} \)
53 \( 1 + (6.03 - 6.03i)T - 53iT^{2} \)
59 \( 1 + (-5.50 + 1.47i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.64 - 4.58i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.66 + 2.69i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.0 - 2.95i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 5.27iT - 73T^{2} \)
79 \( 1 - 4.66iT - 79T^{2} \)
83 \( 1 - 0.863T + 83T^{2} \)
89 \( 1 + (-3.59 + 13.4i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-15.4 - 8.93i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.25400154438976493422626200702, −9.956652362749342463056258511900, −9.114622254633225586953382536436, −8.736989215467924240239181180813, −7.33183179430479806909486684455, −6.36788228315842850986320123096, −5.53409350534141568950729785175, −4.50584508943482893070978897334, −2.49610627206295452676774690115, −0.66251158619757081263477840341, 1.50005697788257949315816997977, 3.52446381431592051465974289861, 4.07507847559208165149452158935, 6.24438888967346761978428153757, 6.67174001394155436365267124999, 7.59032329166771871375464540751, 9.003190821047877794294532708041, 9.864029654973386906104943234912, 10.58671441542891930733189733407, 11.24298022563686024567516477638

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