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Properties

Label 2-630-63.58-c1-0-31
Degree $2$
Conductor $630$
Sign $-0.0177 + 0.999i$
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 + (0.451 − 1.67i)3-s + 4-s + (−0.5 − 0.866i)5-s + (0.451 − 1.67i)6-s + (1.85 − 1.88i)7-s + 8-s + (−2.59 − 1.51i)9-s + (−0.5 − 0.866i)10-s + (−1.39 + 2.41i)11-s + (0.451 − 1.67i)12-s + (0.900 − 1.55i)13-s + (1.85 − 1.88i)14-s + (−1.67 + 0.444i)15-s + 16-s + (−0.292 − 0.506i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.260 − 0.965i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (0.184 − 0.682i)6-s + (0.699 − 0.714i)7-s + 0.353·8-s + (−0.864 − 0.503i)9-s + (−0.158 − 0.273i)10-s + (−0.420 + 0.727i)11-s + (0.130 − 0.482i)12-s + (0.249 − 0.432i)13-s + (0.494 − 0.505i)14-s + (−0.432 + 0.114i)15-s + 0.250·16-s + (−0.0708 − 0.122i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0177 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0177 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0177 + 0.999i$
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: \((2,\ 630,\ (\ :1/2),\ -0.0177 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68695 - 1.71713i\)
\(L(\frac12)\) \(\approx\) \(1.68695 - 1.71713i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.451 + 1.67i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.85 + 1.88i)T \)
good11 \( 1 + (1.39 - 2.41i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.900 + 1.55i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.292 + 0.506i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.54 + 4.40i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.60 - 2.78i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.226 - 0.392i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.90T + 31T^{2} \)
37 \( 1 + (-1.10 + 1.91i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.43 - 4.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.41 - 4.17i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + (-2.04 - 3.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 7.99T + 59T^{2} \)
61 \( 1 - 15.1T + 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 + 5.78T + 71T^{2} \)
73 \( 1 + (-3.03 - 5.26i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 0.652T + 79T^{2} \)
83 \( 1 + (-1.43 - 2.48i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.58 + 2.74i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.00 - 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.70093413398408289885889111365, −9.407000208679725398379161072992, −8.375435405340773213371346602888, −7.39815084781736851130335572886, −7.13664514531278793513089724250, −5.70057833558100753947842327938, −4.87924943253007301637798915508, −3.71755748061334987230041504660, −2.43490723245316537544912732641, −1.10951902471592919126009455611, 2.21053574335449866857004166403, 3.32899745512400441925194814122, 4.19705197333073755836075466306, 5.34786076425923791333339877138, 5.87148245924470087984500457821, 7.30247126063037504777481347355, 8.326157732302907362700943072961, 8.966625538613697593116316489121, 10.18576092232862845226548282612, 10.91139897349222390889410690424

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