Properties

Label 2-390-13.4-c1-0-4
Degree $2$
Conductor $390$
Sign $0.472 + 0.881i$
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

Related objects

Downloads

Learn more

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 + (1.14 − 0.661i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−3.99 − 2.30i)11-s − 0.999·12-s + (3.20 − 1.66i)13-s − 1.32·14-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 + (0.432 − 0.249i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (−1.20 − 0.695i)11-s − 0.288·12-s + (0.887 − 0.460i)13-s − 0.353·14-s + (0.223 + 0.129i)15-s + (−0.125 + 0.216i)16-s + (0.485 + 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.472 + 0.881i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.472 + 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.765070 - 0.458149i\)
\(L(\frac12)\) \(\approx\) \(0.765070 - 0.458149i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.20 + 1.66i)T \)
good7 \( 1 + (-1.14 + 0.661i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.99 + 2.30i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.98 + 1.14i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.33 + 7.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.01 + 1.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.1iT - 31T^{2} \)
37 \( 1 + (-5.89 - 3.40i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.02 + 2.32i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.30 - 7.45i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.10iT - 47T^{2} \)
53 \( 1 - 0.826T + 53T^{2} \)
59 \( 1 + (-2.72 + 1.57i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.267 + 0.464i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.75 + 1.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.81 - 5.66i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.28iT - 73T^{2} \)
79 \( 1 - 2.96T + 79T^{2} \)
83 \( 1 - 15.8iT - 83T^{2} \)
89 \( 1 + (-10.2 - 5.93i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.63 - 4.40i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.88845152770361883651574638383, −10.46644083533281458212037885412, −9.405263274352364336225242798594, −8.319639215697678872049437434089, −7.910416588552526009968595050812, −6.29674847757266891376947317076, −5.28989274472948888512483271000, −4.10136917398346176645609329790, −2.77713830057797031774245870276, −0.823603500455178174541225069049, 1.52085986021617002388305811002, 3.01446379869990457811133566874, 4.98215643761354715274475957794, 5.78288673553626082121029680709, 7.08653624044825642765728296819, 7.54124431534198846082319116549, 8.604112063264459874367879774606, 9.620967721909921180981514834625, 10.60672188090926765814798173708, 11.32568075256082586398095102605

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