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Properties

Label 2-630-35.33-c1-0-13
Degree $2$
Conductor $630$
Sign $-0.0172 + 0.999i$
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  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.774 − 2.09i)5-s + (2.64 − 0.126i)7-s + (0.707 + 0.707i)8-s + (−1.82 + 1.29i)10-s + (2.81 + 4.87i)11-s + (1.42 − 1.42i)13-s + (−0.806 − 2.51i)14-s + (0.500 − 0.866i)16-s + (1.37 − 5.12i)17-s + (1.94 − 3.37i)19-s + (1.71 + 1.42i)20-s + (3.97 − 3.97i)22-s + (−1.08 + 0.290i)23-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.346 − 0.938i)5-s + (0.998 − 0.0477i)7-s + (0.249 + 0.249i)8-s + (−0.577 + 0.408i)10-s + (0.848 + 1.46i)11-s + (0.396 − 0.396i)13-s + (−0.215 − 0.673i)14-s + (0.125 − 0.216i)16-s + (0.333 − 1.24i)17-s + (0.446 − 0.773i)19-s + (0.384 + 0.319i)20-s + (0.848 − 0.848i)22-s + (−0.226 + 0.0606i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0172 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0172 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0172 + 0.999i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.0172 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.985064 - 1.00216i\)
\(L(\frac12)\) \(\approx\) \(0.985064 - 1.00216i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 \)
5 \( 1 + (0.774 + 2.09i)T \)
7 \( 1 + (-2.64 + 0.126i)T \)
good11 \( 1 + (-2.81 - 4.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.42 + 1.42i)T - 13iT^{2} \)
17 \( 1 + (-1.37 + 5.12i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.94 + 3.37i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.08 - 0.290i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 3.15iT - 29T^{2} \)
31 \( 1 + (3.33 - 1.92i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.30 + 4.86i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.21iT - 41T^{2} \)
43 \( 1 + (-1.85 - 1.85i)T + 43iT^{2} \)
47 \( 1 + (-5.69 + 1.52i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.357 + 1.33i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.73 - 4.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.99 + 2.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.816 + 0.218i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.77T + 71T^{2} \)
73 \( 1 + (-5.42 - 1.45i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.41 + 3.12i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.67 + 5.67i)T - 83iT^{2} \)
89 \( 1 + (5.96 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.63 - 6.63i)T + 97iT^{2} \)
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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

−10.40493713661763569198280469006, −9.342208485940641554802909650669, −8.941889819723380020863360434103, −7.77627711295113057650529643833, −7.18656524471831663274254430248, −5.41245640177877183644882085618, −4.68134699030956098353025335725, −3.84484106946892002421912132107, −2.16919217681946424645019821924, −0.981609851928948372742980243870, 1.47045975906220467538209455184, 3.37326789717661710567951912375, 4.20859617561997475798694958496, 5.71895348150591538054855325057, 6.26329108023602811459792059204, 7.34135389251959524382764142856, 8.217677378557810328303695363511, 8.717991328484331507905214352921, 9.967428549052866758273818056405, 10.92659252363954936731699653294

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