Properties

Label 2-630-15.8-c1-0-7
Degree $2$
Conductor $630$
Sign $0.636 - 0.771i$
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 + (1.52 − 1.63i)5-s + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (2.23 − 0.0743i)10-s + 2.14i·11-s + (2.16 + 2.16i)13-s − 1.00·14-s − 1.00·16-s + (4.46 + 4.46i)17-s + (1.63 + 1.52i)20-s + (−1.51 + 1.51i)22-s + (6.32 − 6.32i)23-s + (−0.332 − 4.98i)25-s + 3.05i·26-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.683 − 0.730i)5-s + (−0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s + (0.706 − 0.0234i)10-s + 0.647i·11-s + (0.599 + 0.599i)13-s − 0.267·14-s − 0.250·16-s + (1.08 + 1.08i)17-s + (0.365 + 0.341i)20-s + (−0.323 + 0.323i)22-s + (1.31 − 1.31i)23-s + (−0.0664 − 0.997i)25-s + 0.599i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.636 - 0.771i$
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: \((2,\ 630,\ (\ :1/2),\ 0.636 - 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96655 + 0.926464i\)
\(L(\frac12)\) \(\approx\) \(1.96655 + 0.926464i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 + 1.63i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 - 2.14iT - 11T^{2} \)
13 \( 1 + (-2.16 - 2.16i)T + 13iT^{2} \)
17 \( 1 + (-4.46 - 4.46i)T + 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-6.32 + 6.32i)T - 23iT^{2} \)
29 \( 1 - 8.20T + 29T^{2} \)
31 \( 1 + 1.85T + 31T^{2} \)
37 \( 1 + (5.13 - 5.13i)T - 37iT^{2} \)
41 \( 1 - 7.56iT - 41T^{2} \)
43 \( 1 + (7.84 + 7.84i)T + 43iT^{2} \)
47 \( 1 + (5.01 + 5.01i)T + 47iT^{2} \)
53 \( 1 + (-2.35 + 2.35i)T - 53iT^{2} \)
59 \( 1 + 9.35T + 59T^{2} \)
61 \( 1 + 4.46T + 61T^{2} \)
67 \( 1 + (3.84 - 3.84i)T - 67iT^{2} \)
71 \( 1 - 0.420iT - 71T^{2} \)
73 \( 1 + (2.63 + 2.63i)T + 73iT^{2} \)
79 \( 1 - 5.23iT - 79T^{2} \)
83 \( 1 + (-10.6 + 10.6i)T - 83iT^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + (0.606 - 0.606i)T - 97iT^{2} \)
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

−10.56214593898237780765273673193, −9.824349165766300063089883741785, −8.738975896396418102785675448396, −8.289536796354017814407489988858, −6.85340694996879274054286897546, −6.24156692679608233049376092899, −5.20123423674023995272623245995, −4.43970966022175225717561703032, −3.10239416919458910532689596488, −1.58650892883983594001900882867, 1.23643976880131727076912142236, 2.96024461443265103933532551798, 3.40098260984895911252632608663, 5.06119274486268181807176787007, 5.80146093518682869897514758774, 6.74896884141069669881266659081, 7.68287508436169383860986145634, 9.028310182292378202215635591571, 9.793417127706419572912437479808, 10.61500998861415617079104326166

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