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Properties

Label 2-630-63.58-c1-0-8
Degree $2$
Conductor $630$
Sign $0.271 - 0.962i$
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  − 2-s + (−1.70 + 0.314i)3-s + 4-s + (0.5 + 0.866i)5-s + (1.70 − 0.314i)6-s + (2.48 + 0.917i)7-s − 8-s + (2.80 − 1.07i)9-s + (−0.5 − 0.866i)10-s + (−1.14 + 1.98i)11-s + (−1.70 + 0.314i)12-s + (−0.845 + 1.46i)13-s + (−2.48 − 0.917i)14-s + (−1.12 − 1.31i)15-s + 16-s + (0.372 + 0.644i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.983 + 0.181i)3-s + 0.5·4-s + (0.223 + 0.387i)5-s + (0.695 − 0.128i)6-s + (0.937 + 0.346i)7-s − 0.353·8-s + (0.934 − 0.357i)9-s + (−0.158 − 0.273i)10-s + (−0.346 + 0.599i)11-s + (−0.491 + 0.0907i)12-s + (−0.234 + 0.406i)13-s + (−0.663 − 0.245i)14-s + (−0.290 − 0.340i)15-s + 0.250·16-s + (0.0902 + 0.156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.271 - 0.962i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.271 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.659584 + 0.499331i\)
\(L(\frac12)\) \(\approx\) \(0.659584 + 0.499331i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.70 - 0.314i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.48 - 0.917i)T \)
good11 \( 1 + (1.14 - 1.98i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.845 - 1.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.372 - 0.644i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.89 + 6.75i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.07 + 1.86i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.61 - 4.53i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.29T + 31T^{2} \)
37 \( 1 + (5.40 - 9.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.14 - 3.72i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.16 - 7.21i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.18T + 47T^{2} \)
53 \( 1 + (-6.31 - 10.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.635T + 59T^{2} \)
61 \( 1 - 6.90T + 61T^{2} \)
67 \( 1 - 9.77T + 67T^{2} \)
71 \( 1 - 2.50T + 71T^{2} \)
73 \( 1 + (3.42 + 5.93i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 9.30T + 79T^{2} \)
83 \( 1 + (0.645 + 1.11i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.165 + 0.286i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.18 - 5.51i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.72670210421574495665350303648, −10.01847721914743995507845254257, −9.216119743747661721765304794977, −8.135292617157217729223410740635, −7.13304255728079332896293848388, −6.49221203613792312211096549398, −5.24399701875536380864603951065, −4.59835997687783400972930951653, −2.74187744056219114541418189290, −1.34140063108091855632433071203, 0.73193651490986429127265168186, 1.95130698274305457789969861333, 3.85930395095057460464602479585, 5.27083450949289015335100787639, 5.71519918585644469826123914520, 7.00705890041821233290935984915, 7.84677849690183822104040774790, 8.478993142811980575630576237651, 9.893946336115123427243849821382, 10.29150027880580217900045986651

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