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Properties

Label 2-630-5.2-c2-0-11
Degree $2$
Conductor $630$
Sign $0.999 - 0.0260i$
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  + (−1 − i)2-s + 2i·4-s + (−4.32 − 2.51i)5-s + (1.87 + 1.87i)7-s + (2 − 2i)8-s + (1.80 + 6.83i)10-s − 2.92·11-s + (−1.13 + 1.13i)13-s − 3.74i·14-s − 4·16-s + (−1.54 − 1.54i)17-s + 3.35i·19-s + (5.03 − 8.64i)20-s + (2.92 + 2.92i)22-s + (−7.90 + 7.90i)23-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + 0.5i·4-s + (−0.864 − 0.503i)5-s + (0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s + (0.180 + 0.683i)10-s − 0.265·11-s + (−0.0871 + 0.0871i)13-s − 0.267i·14-s − 0.250·16-s + (−0.0908 − 0.0908i)17-s + 0.176i·19-s + (0.251 − 0.432i)20-s + (0.132 + 0.132i)22-s + (−0.343 + 0.343i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0260i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.999 - 0.0260i$
Analytic conductor: \(17.1662\)
Root analytic conductor: \(4.14321\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1),\ 0.999 - 0.0260i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9983455232\)
\(L(\frac12)\) \(\approx\) \(0.9983455232\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 \)
5 \( 1 + (4.32 + 2.51i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good11 \( 1 + 2.92T + 121T^{2} \)
13 \( 1 + (1.13 - 1.13i)T - 169iT^{2} \)
17 \( 1 + (1.54 + 1.54i)T + 289iT^{2} \)
19 \( 1 - 3.35iT - 361T^{2} \)
23 \( 1 + (7.90 - 7.90i)T - 529iT^{2} \)
29 \( 1 - 13.5iT - 841T^{2} \)
31 \( 1 - 15.7T + 961T^{2} \)
37 \( 1 + (16.8 + 16.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 72.6T + 1.68e3T^{2} \)
43 \( 1 + (-20.3 + 20.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (-52.3 - 52.3i)T + 2.20e3iT^{2} \)
53 \( 1 + (-40.5 + 40.5i)T - 2.80e3iT^{2} \)
59 \( 1 - 117. iT - 3.48e3T^{2} \)
61 \( 1 + 45.4T + 3.72e3T^{2} \)
67 \( 1 + (-57.7 - 57.7i)T + 4.48e3iT^{2} \)
71 \( 1 - 51.7T + 5.04e3T^{2} \)
73 \( 1 + (-72.3 + 72.3i)T - 5.32e3iT^{2} \)
79 \( 1 + 37.1iT - 6.24e3T^{2} \)
83 \( 1 + (-21.3 + 21.3i)T - 6.88e3iT^{2} \)
89 \( 1 - 100. iT - 7.92e3T^{2} \)
97 \( 1 + (-52.5 - 52.5i)T + 9.40e3iT^{2} \)
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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

−10.53196096723932719069306749604, −9.389617047917708310740398544820, −8.741904835682406520625876500028, −7.87130015278740255541826324174, −7.21969249062059070335603037296, −5.77248575156058076352409648582, −4.63473909792116901088609831835, −3.70896786716877470012748202088, −2.40920874736065232667774091775, −0.913979485284916062294400127244, 0.60178327405729578614382513391, 2.44200573942726316479799509588, 3.85114126368883268318277566950, 4.84926465390939312681399136544, 6.09422784749818320681779065253, 7.00466379367950690577252359925, 7.78916794037357280317270359305, 8.394696534431998794990302148419, 9.477005677627285585599435049812, 10.44931915488142791745811911573

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