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Properties

Label 2-630-7.3-c2-0-23
Degree $2$
Conductor $630$
Sign $0.407 + 0.913i$
Analytic cond. $17.1662$
Root an. cond. $4.14321$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 1.22i)2-s + (−0.999 + 1.73i)4-s + (1.93 − 1.11i)5-s + (−5.10 − 4.79i)7-s − 2.82·8-s + (2.73 + 1.58i)10-s + (−9.03 + 15.6i)11-s − 18.6i·13-s + (2.26 − 9.63i)14-s + (−2.00 − 3.46i)16-s + (−1.33 − 0.770i)17-s + (29.4 − 17.0i)19-s + 4.47i·20-s − 25.5·22-s + (−13.4 − 23.3i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (−0.728 − 0.684i)7-s − 0.353·8-s + (0.273 + 0.158i)10-s + (−0.821 + 1.42i)11-s − 1.43i·13-s + (0.161 − 0.688i)14-s + (−0.125 − 0.216i)16-s + (−0.0785 − 0.0453i)17-s + (1.55 − 0.895i)19-s + 0.223i·20-s − 1.16·22-s + (−0.586 − 1.01i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.407 + 0.913i$
Analytic conductor: \(17.1662\)
Root analytic conductor: \(4.14321\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1),\ 0.407 + 0.913i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.290288040\)
\(L(\frac12)\) \(\approx\) \(1.290288040\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (5.10 + 4.79i)T \)
good11 \( 1 + (9.03 - 15.6i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 18.6iT - 169T^{2} \)
17 \( 1 + (1.33 + 0.770i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-29.4 + 17.0i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (13.4 + 23.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 16.4T + 841T^{2} \)
31 \( 1 + (24.1 + 13.9i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (25.8 + 44.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 37.4iT - 1.68e3T^{2} \)
43 \( 1 + 63.6T + 1.84e3T^{2} \)
47 \( 1 + (-24.4 + 14.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-0.221 + 0.384i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (64.2 + 37.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-91.0 + 52.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-6.35 + 11.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 45.7T + 5.04e3T^{2} \)
73 \( 1 + (-31.5 - 18.2i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-66.5 - 115. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 49.9iT - 6.88e3T^{2} \)
89 \( 1 + (85.9 - 49.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 150. iT - 9.40e3T^{2} \)
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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

−10.04437301190285199944326567088, −9.534574386311877599780372590814, −8.263976182725297112796160615220, −7.39208880786446948039802028848, −6.79858368509415427955788220349, −5.52272617533709084590557174846, −4.91647896566104343443528524648, −3.64975498269709834723372149565, −2.48003899883849490757146405178, −0.41355896970281087137682904852, 1.55911381183855691757920152119, 2.91151327300464044594860437931, 3.59879803432869415016879837411, 5.17321536716475145264069357657, 5.85796684231044582365541844535, 6.73232447420763901576813803154, 8.082497499440895329361551447142, 9.068895401491687532388785256926, 9.753584471631905129369291426076, 10.53276878403614675330221643008

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