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Properties

Label 2-630-315.158-c1-0-24
Degree $2$
Conductor $630$
Sign $0.512 + 0.858i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.754 + 1.55i)3-s + (0.866 − 0.499i)4-s + (−1.96 − 1.07i)5-s + (−1.13 − 1.31i)6-s + (−2.51 + 0.824i)7-s + (−0.707 + 0.707i)8-s + (−1.86 + 2.35i)9-s + (2.17 + 0.525i)10-s − 4.53i·11-s + (1.43 + 0.972i)12-s + (−3.26 + 0.873i)13-s + (2.21 − 1.44i)14-s + (0.187 − 3.86i)15-s + (0.500 − 0.866i)16-s + (6.71 − 1.79i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.435 + 0.900i)3-s + (0.433 − 0.249i)4-s + (−0.878 − 0.478i)5-s + (−0.462 − 0.535i)6-s + (−0.950 + 0.311i)7-s + (−0.249 + 0.249i)8-s + (−0.620 + 0.784i)9-s + (0.687 + 0.166i)10-s − 1.36i·11-s + (0.413 + 0.280i)12-s + (−0.904 + 0.242i)13-s + (0.592 − 0.386i)14-s + (0.0483 − 0.998i)15-s + (0.125 − 0.216i)16-s + (1.62 − 0.436i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.512 + 0.858i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (473, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.512 + 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.528078 - 0.299619i\)
\(L(\frac12)\) \(\approx\) \(0.528078 - 0.299619i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.754 - 1.55i)T \)
5 \( 1 + (1.96 + 1.07i)T \)
7 \( 1 + (2.51 - 0.824i)T \)
good11 \( 1 + 4.53iT - 11T^{2} \)
13 \( 1 + (3.26 - 0.873i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-6.71 + 1.79i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.85 + 2.80i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.83 + 3.83i)T - 23iT^{2} \)
29 \( 1 + (4.57 + 7.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.880 + 1.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.47 + 1.73i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.832 + 0.480i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.841 - 3.13i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.09 - 0.293i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.424 + 1.58i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.00 + 6.94i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.09 + 3.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.25 + 2.47i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 12.8iT - 71T^{2} \)
73 \( 1 + (-8.08 + 2.16i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.30 + 3.06i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.33 + 1.42i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (7.21 + 12.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.0 - 3.23i)T + (84.0 + 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.14958127408607157207456448226, −9.469070969468310862681287894912, −8.869239884088977500119547488837, −7.987972912742454576815082756584, −7.23168004510262781679724766682, −5.80285284004937018447235515662, −4.98548029511093182324356444194, −3.54240657188370973105001284543, −2.87793890598885845485708580938, −0.41159160504471230168917924943, 1.41689953897093848727991341291, 3.00540654577395668643283802335, 3.58320955488422343220215997396, 5.43237697720519494716667865089, 6.87259571114785109170565342430, 7.36479173526248572897037759254, 7.76697673079364822913666742273, 9.058578332121585492111575045259, 9.850370628783952179678956407773, 10.51047410724761979634171219820

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