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Properties

Label 2-630-7.2-c1-0-3
Degree $2$
Conductor $630$
Sign $0.895 + 0.444i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 2.64·7-s − 0.999·8-s + (0.499 + 0.866i)10-s + (2.32 + 4.02i)11-s + 0.645·13-s + (1.32 − 2.29i)14-s + (−0.5 + 0.866i)16-s + (1.64 + 2.85i)17-s + (2.14 − 3.71i)19-s + 0.999·20-s + 4.64·22-s + (1.5 − 2.59i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 0.999·7-s − 0.353·8-s + (0.158 + 0.273i)10-s + (0.700 + 1.21i)11-s + 0.179·13-s + (0.353 − 0.612i)14-s + (−0.125 + 0.216i)16-s + (0.399 + 0.691i)17-s + (0.492 − 0.852i)19-s + 0.223·20-s + 0.990·22-s + (0.312 − 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88062 - 0.440557i\)
\(L(\frac12)\) \(\approx\) \(1.88062 - 0.440557i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 - 2.64T \)
good11 \( 1 + (-2.32 - 4.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.645T + 13T^{2} \)
17 \( 1 + (-1.64 - 2.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.14 + 3.71i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.96 + 5.14i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.35T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + (4.79 - 8.29i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.145 + 0.252i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.64 + 8.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.64 - 9.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.35 + 4.07i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.70T + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.64 - 9.77i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + (-3.29 + 5.70i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−10.81425332046294388938296738044, −9.763345391755953035314229321404, −8.977223526869068041802925428075, −7.84495816390185142929186156570, −7.02404575623162728674459159839, −5.86809985297340354358007695655, −4.69911878429466426706604232578, −4.03539013202393980848623522385, −2.62691098707365688650859915755, −1.41773572250453284488507145049, 1.23421690309365059776694844834, 3.19674404407841627328342028108, 4.22120841438400863230902083749, 5.27201720699395148238851637895, 5.96902246016663420650824861868, 7.19677243180717480241731820263, 8.025410930827953578766960567633, 8.684284326639562805544883034215, 9.556682097353795246374329790681, 10.86808591716350716838040818027

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