↑

Properties

Label 2-930-155.149-c1-0-23
Degree $2$
Conductor $930$
Sign $-0.975 - 0.221i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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  − i·2-s + (−0.866 + 0.5i)3-s − 4-s + (−1.51 + 1.64i)5-s + (0.5 + 0.866i)6-s + (1.77 − 1.02i)7-s + i·8-s + (0.499 − 0.866i)9-s + (1.64 + 1.51i)10-s + (−1.58 + 2.74i)11-s + (0.866 − 0.5i)12-s + (−0.551 − 0.318i)13-s + (−1.02 − 1.77i)14-s + (0.485 − 2.18i)15-s + 16-s + (0.148 − 0.0855i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.499 + 0.288i)3-s − 0.5·4-s + (−0.676 + 0.736i)5-s + (0.204 + 0.353i)6-s + (0.670 − 0.387i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.520 + 0.478i)10-s + (−0.477 + 0.826i)11-s + (0.249 − 0.144i)12-s + (−0.152 − 0.0882i)13-s + (−0.273 − 0.474i)14-s + (0.125 − 0.563i)15-s + 0.250·16-s + (0.0359 − 0.0207i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.975 - 0.221i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.975 - 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00559350 + 0.0499484i\)
\(L(\frac12)\) \(\approx\) \(0.00559350 + 0.0499484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (1.51 - 1.64i)T \)
31 \( 1 + (5.11 + 2.19i)T \)
good7 \( 1 + (-1.77 + 1.02i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.58 - 2.74i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.551 + 0.318i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.148 + 0.0855i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.32 + 2.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.77iT - 23T^{2} \)
29 \( 1 + 8.42T + 29T^{2} \)
37 \( 1 + (-3.31 + 1.91i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.24 - 2.15i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.07 - 1.77i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.09iT - 47T^{2} \)
53 \( 1 + (8.61 + 4.97i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.86 - 3.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 + (4.79 + 2.76i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.97 - 8.61i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.44 + 4.29i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.97 + 8.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.653 + 0.377i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 + 16.4iT - 97T^{2} \)
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

−9.915571495974350899596854716493, −8.976440273136958317413015046813, −7.75776422944110576565628584215, −7.31372176699277629463526474980, −6.08121892312465781549342627069, −4.86125467688014773929762839020, −4.26417217950738984996426149294, −3.17516581894959954740783622205, −1.90563812047288914800632295390, −0.02524168771037916344730438524, 1.60417739562049002511178094162, 3.49043948251069054432687816536, 4.62727957579675526575829327345, 5.39777288338411887541532567804, 6.04520389448948751629118511902, 7.34393833371923998588930821014, 7.894228320964349026762732094200, 8.635646930480328248334041020460, 9.389702480864857890683735547844, 10.67008620071002108591360300981

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