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Properties

Label 2-630-63.25-c1-0-30
Degree $2$
Conductor $630$
Sign $-0.990 + 0.139i$
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 + (0.0520 − 1.73i)3-s + 4-s + (0.5 − 0.866i)5-s + (−0.0520 + 1.73i)6-s + (0.226 − 2.63i)7-s − 8-s + (−2.99 − 0.180i)9-s + (−0.5 + 0.866i)10-s + (−2.57 − 4.45i)11-s + (0.0520 − 1.73i)12-s + (1.62 + 2.81i)13-s + (−0.226 + 2.63i)14-s + (−1.47 − 0.910i)15-s + 16-s + (−2.89 + 5.01i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.0300 − 0.999i)3-s + 0.5·4-s + (0.223 − 0.387i)5-s + (−0.0212 + 0.706i)6-s + (0.0856 − 0.996i)7-s − 0.353·8-s + (−0.998 − 0.0601i)9-s + (−0.158 + 0.273i)10-s + (−0.775 − 1.34i)11-s + (0.0150 − 0.499i)12-s + (0.450 + 0.779i)13-s + (−0.0605 + 0.704i)14-s + (−0.380 − 0.235i)15-s + 0.250·16-s + (−0.702 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.990 + 0.139i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.990 + 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0520313 - 0.743021i\)
\(L(\frac12)\) \(\approx\) \(0.0520313 - 0.743021i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0520 + 1.73i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.226 + 2.63i)T \)
good11 \( 1 + (2.57 + 4.45i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.62 - 2.81i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.89 - 5.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.34 + 4.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.33 + 2.31i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.227 + 0.394i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.14T + 31T^{2} \)
37 \( 1 + (-2.00 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.71 + 6.42i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.80 - 3.12i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.15T + 47T^{2} \)
53 \( 1 + (1.63 - 2.82i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 - 5.99T + 61T^{2} \)
67 \( 1 + 11.7T + 67T^{2} \)
71 \( 1 + 4.38T + 71T^{2} \)
73 \( 1 + (-6.94 + 12.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 2.95T + 79T^{2} \)
83 \( 1 + (-6.13 + 10.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.83 + 8.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.91 + 6.78i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.37782403566703867192471437547, −8.928575420481612075140675955261, −8.474463934309499359584352340524, −7.69943779808146862889903482360, −6.57168048470446678511042706421, −6.13301856755540501086194312419, −4.60851409487746359945791476466, −3.10168840477517416971764563601, −1.74492879988632525760616594683, −0.48967435316127325420660519437, 2.22940446256406910489669197778, 3.09551448178640134109120523131, 4.67259988780745749858699721357, 5.51955034829734599438611257499, 6.52968911226036577755446030773, 7.77131475701518352519224205686, 8.520286283305729574271514064106, 9.497355588483840981358965791470, 9.979422486370781059093671693161, 10.76888515532100329050403365301

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