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Properties

Label 2-630-5.2-c2-0-29
Degree $2$
Conductor $630$
Sign $0.0972 + 0.995i$
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 + (2.20 − 4.48i)5-s + (−1.87 − 1.87i)7-s + (−2 + 2i)8-s + (6.69 − 2.28i)10-s − 15.1·11-s + (14.6 − 14.6i)13-s − 3.74i·14-s − 4·16-s + (−15.9 − 15.9i)17-s + 2.05i·19-s + (8.97 + 4.40i)20-s + (−15.1 − 15.1i)22-s + (−19.4 + 19.4i)23-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + (0.440 − 0.897i)5-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s + (0.669 − 0.228i)10-s − 1.37·11-s + (1.12 − 1.12i)13-s − 0.267i·14-s − 0.250·16-s + (−0.940 − 0.940i)17-s + 0.107i·19-s + (0.448 + 0.220i)20-s + (−0.687 − 0.687i)22-s + (−0.847 + 0.847i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0972 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0972 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.0972 + 0.995i$
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: \((2,\ 630,\ (\ :1),\ 0.0972 + 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.597589460\)
\(L(\frac12)\) \(\approx\) \(1.597589460\)
\(L(2)\) 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 \( 1 + (-2.20 + 4.48i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good11 \( 1 + 15.1T + 121T^{2} \)
13 \( 1 + (-14.6 + 14.6i)T - 169iT^{2} \)
17 \( 1 + (15.9 + 15.9i)T + 289iT^{2} \)
19 \( 1 - 2.05iT - 361T^{2} \)
23 \( 1 + (19.4 - 19.4i)T - 529iT^{2} \)
29 \( 1 + 33.1iT - 841T^{2} \)
31 \( 1 - 27.8T + 961T^{2} \)
37 \( 1 + (30.4 + 30.4i)T + 1.36e3iT^{2} \)
41 \( 1 - 26.6T + 1.68e3T^{2} \)
43 \( 1 + (-7.89 + 7.89i)T - 1.84e3iT^{2} \)
47 \( 1 + (33.8 + 33.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (5.60 - 5.60i)T - 2.80e3iT^{2} \)
59 \( 1 + 5.80iT - 3.48e3T^{2} \)
61 \( 1 + 98.2T + 3.72e3T^{2} \)
67 \( 1 + (-51.6 - 51.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 120.T + 5.04e3T^{2} \)
73 \( 1 + (-81.9 + 81.9i)T - 5.32e3iT^{2} \)
79 \( 1 - 33.0iT - 6.24e3T^{2} \)
83 \( 1 + (-97.0 + 97.0i)T - 6.88e3iT^{2} \)
89 \( 1 - 34.2iT - 7.92e3T^{2} \)
97 \( 1 + (-104. - 104. i)T + 9.40e3iT^{2} \)
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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

−10.15679935411308903524373887817, −9.232317320863832305571438231371, −8.207761171956189518063695363788, −7.70094422354478252293326197401, −6.34863755144194443163404381381, −5.57635888822726911910287925714, −4.82662306022568740468402750203, −3.64940818015524919471923901342, −2.34710964307425724665756660501, −0.46527423705379565782956124462, 1.83654819177214864732265808583, 2.77363868067898399726154403558, 3.86590508468933360358395373865, 5.02054569562281244196985919835, 6.26221409791183085311746676606, 6.57246951809009423813181920996, 8.035550132928799794692216579066, 8.982438573766659459414471651887, 10.01272456963590098309006663216, 10.76479444733984527039324706844

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