↑

Properties

Label 2-630-105.2-c1-0-13
Degree $2$
Conductor $630$
Sign $0.658 + 0.752i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (1.24 − 1.85i)5-s + (−1.31 − 2.29i)7-s + (−0.707 − 0.707i)8-s + (2.11 + 0.719i)10-s + (−1.05 + 0.608i)11-s + (2.42 − 2.42i)13-s + (1.87 − 1.86i)14-s + (0.500 − 0.866i)16-s + (−3.34 − 0.896i)17-s + (−7.32 − 4.22i)19-s + (−0.146 + 2.23i)20-s + (−0.859 − 0.859i)22-s + (7.15 − 1.91i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (0.555 − 0.831i)5-s + (−0.497 − 0.867i)7-s + (−0.249 − 0.249i)8-s + (0.669 + 0.227i)10-s + (−0.317 + 0.183i)11-s + (0.672 − 0.672i)13-s + (0.501 − 0.498i)14-s + (0.125 − 0.216i)16-s + (−0.811 − 0.217i)17-s + (−1.68 − 0.970i)19-s + (−0.0328 + 0.498i)20-s + (−0.183 − 0.183i)22-s + (1.49 − 0.399i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.658 + 0.752i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.658 + 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24404 - 0.564619i\)
\(L(\frac12)\) \(\approx\) \(1.24404 - 0.564619i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 \)
5 \( 1 + (-1.24 + 1.85i)T \)
7 \( 1 + (1.31 + 2.29i)T \)
good11 \( 1 + (1.05 - 0.608i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.42 + 2.42i)T - 13iT^{2} \)
17 \( 1 + (3.34 + 0.896i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (7.32 + 4.22i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.15 + 1.91i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 7.86T + 29T^{2} \)
31 \( 1 + (3.47 + 6.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.19 + 1.12i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.75iT - 41T^{2} \)
43 \( 1 + (-1.25 + 1.25i)T - 43iT^{2} \)
47 \( 1 + (-1.94 - 7.27i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.15 - 8.03i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.0764 + 0.132i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.62 + 11.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.50 - 13.0i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + (-7.69 - 2.06i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.09 + 0.634i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.08 + 3.08i)T + 83iT^{2} \)
89 \( 1 + (-1.92 + 3.33i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.22 - 3.22i)T + 97iT^{2} \)
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

−10.46043546315613936279139844007, −9.381151877880128777996136281733, −8.734362373662271111346223416804, −7.84694384014001892416212564816, −6.71181821234954931269539532576, −6.12976374773569547658296014327, −4.85332213235941918094390352019, −4.25667986797788627453864330977, −2.69213608275612661906367460617, −0.70541146653281594000147538678, 1.88817945989995555922519683689, 2.82609744310415379044528630508, 3.88749040326701237664908085938, 5.24091222281698819913298720245, 6.24547805533869385239268058028, 6.82031210379630651180573055810, 8.505557607410591813743342447673, 8.992596137964159846365299968861, 10.06533348662181553323099822401, 10.73934071528736915795497825659

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