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Properties

Label 2-630-5.2-c2-0-6
Degree $2$
Conductor $630$
Sign $-0.999 - 0.0327i$
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 + (4.16 + 2.76i)5-s + (−1.87 − 1.87i)7-s + (−2 + 2i)8-s + (1.39 + 6.93i)10-s − 17.6·11-s + (−12.3 + 12.3i)13-s − 3.74i·14-s − 4·16-s + (−18.7 − 18.7i)17-s + 25.5i·19-s + (−5.53 + 8.32i)20-s + (−17.6 − 17.6i)22-s + (5.90 − 5.90i)23-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + (0.832 + 0.553i)5-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s + (0.139 + 0.693i)10-s − 1.60·11-s + (−0.953 + 0.953i)13-s − 0.267i·14-s − 0.250·16-s + (−1.10 − 1.10i)17-s + 1.34i·19-s + (−0.276 + 0.416i)20-s + (−0.803 − 0.803i)22-s + (0.256 − 0.256i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0327i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.999 - 0.0327i$
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: \((2,\ 630,\ (\ :1),\ -0.999 - 0.0327i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.216179312\)
\(L(\frac12)\) \(\approx\) \(1.216179312\)
\(L(2)\) 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 \( 1 + (-4.16 - 2.76i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good11 \( 1 + 17.6T + 121T^{2} \)
13 \( 1 + (12.3 - 12.3i)T - 169iT^{2} \)
17 \( 1 + (18.7 + 18.7i)T + 289iT^{2} \)
19 \( 1 - 25.5iT - 361T^{2} \)
23 \( 1 + (-5.90 + 5.90i)T - 529iT^{2} \)
29 \( 1 - 15.3iT - 841T^{2} \)
31 \( 1 - 11.3T + 961T^{2} \)
37 \( 1 + (-25.5 - 25.5i)T + 1.36e3iT^{2} \)
41 \( 1 + 58.8T + 1.68e3T^{2} \)
43 \( 1 + (-0.282 + 0.282i)T - 1.84e3iT^{2} \)
47 \( 1 + (-48.6 - 48.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (3.93 - 3.93i)T - 2.80e3iT^{2} \)
59 \( 1 + 86.2iT - 3.48e3T^{2} \)
61 \( 1 - 29.2T + 3.72e3T^{2} \)
67 \( 1 + (65.7 + 65.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 1.19T + 5.04e3T^{2} \)
73 \( 1 + (87.0 - 87.0i)T - 5.32e3iT^{2} \)
79 \( 1 - 55.0iT - 6.24e3T^{2} \)
83 \( 1 + (-91.8 + 91.8i)T - 6.88e3iT^{2} \)
89 \( 1 - 103. iT - 7.92e3T^{2} \)
97 \( 1 + (-87.4 - 87.4i)T + 9.40e3iT^{2} \)
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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

−10.70790752954335944137004746794, −9.979888700451594633785853933564, −9.163010487228053111789724559654, −7.941735997999022846344444186670, −7.09799977489658670260921319511, −6.41223375587534114067741060733, −5.32740910693320962550586297451, −4.58212383837380958969287004625, −3.05542240123004020422441640729, −2.19083256775777627280926106314, 0.33523611486158439805929058509, 2.18199546847478952779752805938, 2.84494443826472511016307519542, 4.51932890859853017371361397423, 5.27374910897011683440154684295, 5.99446366111095555193964241073, 7.21641853522772515921251618572, 8.395134916408734394240483014384, 9.209876001888215010275819250488, 10.27718315263367567444605951651

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