Properties

Label 2-630-15.8-c1-0-5
Degree $2$
Conductor $630$
Sign $-0.278 - 0.960i$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.489 + 2.18i)5-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (−1.19 + 1.88i)10-s + 1.77i·11-s + (0.692 + 0.692i)13-s + 1.00·14-s − 1.00·16-s + (2.39 + 2.39i)17-s + (−2.18 + 0.489i)20-s + (−1.25 + 1.25i)22-s + (−3.38 + 3.38i)23-s + (−4.52 + 2.13i)25-s + 0.979i·26-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.218 + 0.975i)5-s + (0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s + (−0.378 + 0.597i)10-s + 0.536i·11-s + (0.192 + 0.192i)13-s + 0.267·14-s − 0.250·16-s + (0.580 + 0.580i)17-s + (−0.487 + 0.109i)20-s + (−0.268 + 0.268i)22-s + (−0.705 + 0.705i)23-s + (−0.904 + 0.427i)25-s + 0.192i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.278 - 0.960i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.278 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17373 + 1.56306i\)
\(L(\frac12)\) \(\approx\) \(1.17373 + 1.56306i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (-0.489 - 2.18i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 - 1.77iT - 11T^{2} \)
13 \( 1 + (-0.692 - 0.692i)T + 13iT^{2} \)
17 \( 1 + (-2.39 - 2.39i)T + 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (3.38 - 3.38i)T - 23iT^{2} \)
29 \( 1 - 4.42T + 29T^{2} \)
31 \( 1 + 5.77T + 31T^{2} \)
37 \( 1 + (-5.91 + 5.91i)T - 37iT^{2} \)
41 \( 1 + 0.807iT - 41T^{2} \)
43 \( 1 + (4.64 + 4.64i)T + 43iT^{2} \)
47 \( 1 + (-7.47 - 7.47i)T + 47iT^{2} \)
53 \( 1 + (3.56 - 3.56i)T - 53iT^{2} \)
59 \( 1 - 5.89T + 59T^{2} \)
61 \( 1 - 2.39T + 61T^{2} \)
67 \( 1 + (0.641 - 0.641i)T - 67iT^{2} \)
71 \( 1 + 10.6iT - 71T^{2} \)
73 \( 1 + (-5.70 - 5.70i)T + 73iT^{2} \)
79 \( 1 + 16.3iT - 79T^{2} \)
83 \( 1 + (-0.171 + 0.171i)T - 83iT^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + (-12.6 + 12.6i)T - 97iT^{2} \)
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

−10.84089473689451321988500197772, −10.08926521483540480286638581159, −9.107625280700442414962519403250, −7.85192088818097206804196437284, −7.31570269424455716840439194606, −6.32218640323725288618269256081, −5.56245823870492659248820353523, −4.28172484281469495936181980860, −3.37021990011501530355805215957, −2.00717971249306768499807238979, 0.968447508978021015991705069809, 2.38525759269725103144732381289, 3.70991439583844926618068227685, 4.81943201014296890080476920453, 5.53283514156194697292255354003, 6.47127821317028424118608932521, 7.931588271283740476686399231496, 8.646950847089125534336555153727, 9.576030806473024286969734119144, 10.34516039817231120650618574808

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