↑

Properties

Label 2-630-35.13-c1-0-2
Degree $2$
Conductor $630$
Sign $-0.836 - 0.548i$
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 + (−1.19 − 1.88i)5-s + (0.510 + 2.59i)7-s + (0.707 + 0.707i)8-s + (2.18 + 0.489i)10-s − 4.79·11-s + (0.585 − 0.585i)13-s + (−2.19 − 1.47i)14-s − 1.00·16-s + (4.10 + 4.10i)17-s − 2.36·19-s + (−1.88 + 1.19i)20-s + (3.39 − 3.39i)22-s + (−2.97 − 2.97i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.535 − 0.844i)5-s + (0.192 + 0.981i)7-s + (0.250 + 0.250i)8-s + (0.689 + 0.154i)10-s − 1.44·11-s + (0.162 − 0.162i)13-s + (−0.587 − 0.394i)14-s − 0.250·16-s + (0.995 + 0.995i)17-s − 0.542·19-s + (−0.422 + 0.267i)20-s + (0.723 − 0.723i)22-s + (−0.621 − 0.621i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.836 - 0.548i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.836 - 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.134202 + 0.449617i\)
\(L(\frac12)\) \(\approx\) \(0.134202 + 0.449617i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 + 1.88i)T \)
7 \( 1 + (-0.510 - 2.59i)T \)
good11 \( 1 + 4.79T + 11T^{2} \)
13 \( 1 + (-0.585 + 0.585i)T - 13iT^{2} \)
17 \( 1 + (-4.10 - 4.10i)T + 17iT^{2} \)
19 \( 1 + 2.36T + 19T^{2} \)
23 \( 1 + (2.97 + 2.97i)T + 23iT^{2} \)
29 \( 1 - 9.94iT - 29T^{2} \)
31 \( 1 - 3.02iT - 31T^{2} \)
37 \( 1 + (6.10 - 6.10i)T - 37iT^{2} \)
41 \( 1 - 10.9iT - 41T^{2} \)
43 \( 1 + (5.74 + 5.74i)T + 43iT^{2} \)
47 \( 1 + (0.363 + 0.363i)T + 47iT^{2} \)
53 \( 1 + (2.36 + 2.36i)T + 53iT^{2} \)
59 \( 1 - 2.07T + 59T^{2} \)
61 \( 1 + 5.55iT - 61T^{2} \)
67 \( 1 + (0.979 - 0.979i)T - 67iT^{2} \)
71 \( 1 + 5.25T + 71T^{2} \)
73 \( 1 + (-6.11 + 6.11i)T - 73iT^{2} \)
79 \( 1 + 5.10iT - 79T^{2} \)
83 \( 1 + (-3.22 + 3.22i)T - 83iT^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + (-8.05 - 8.05i)T + 97iT^{2} \)
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

−10.71295715585345422207657714657, −10.07934041487822166317604349741, −8.869493292036347483842109520431, −8.306074129937388766375510501772, −7.81934246124803967771722603931, −6.47254952255690234663450832133, −5.39543607158273428601695818095, −4.88374433005140513357062585777, −3.25859550172253433226256542885, −1.66405441867546583516164266230, 0.29522668069004027955878630690, 2.27294203162021120141116723756, 3.38930301047317147429043280738, 4.33301799750897758156348347869, 5.69985391282814423852758583210, 7.09471884837089643598314252708, 7.64474399427563733992859358043, 8.282471751177999031215590715837, 9.768401006249681578703205911405, 10.24304562016048546785000340765

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