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Properties

Label 2-630-105.59-c1-0-5
Degree $2$
Conductor $630$
Sign $-0.115 - 0.993i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.98 + 1.02i)5-s + (1.39 + 2.24i)7-s − 0.999·8-s + (0.104 + 2.23i)10-s + (1.37 + 0.796i)11-s + 0.925·13-s + (−1.24 + 2.33i)14-s + (−0.5 − 0.866i)16-s + (−3.42 − 1.97i)17-s + (0.541 − 0.312i)19-s + (−1.88 + 1.20i)20-s + 1.59i·22-s + (−3.89 − 6.74i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.888 + 0.458i)5-s + (0.528 + 0.848i)7-s − 0.353·8-s + (0.0330 + 0.706i)10-s + (0.415 + 0.240i)11-s + 0.256·13-s + (−0.332 + 0.623i)14-s + (−0.125 − 0.216i)16-s + (−0.831 − 0.479i)17-s + (0.124 − 0.0717i)19-s + (−0.420 + 0.269i)20-s + 0.339i·22-s + (−0.811 − 1.40i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.115 - 0.993i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.115 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38166 + 1.55098i\)
\(L(\frac12)\) \(\approx\) \(1.38166 + 1.55098i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-1.98 - 1.02i)T \)
7 \( 1 + (-1.39 - 2.24i)T \)
good11 \( 1 + (-1.37 - 0.796i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.925T + 13T^{2} \)
17 \( 1 + (3.42 + 1.97i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.541 + 0.312i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.89 + 6.74i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 9.34iT - 29T^{2} \)
31 \( 1 + (-8.94 - 5.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.369 - 0.213i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.35T + 41T^{2} \)
43 \( 1 + 6.27iT - 43T^{2} \)
47 \( 1 + (2.40 - 1.39i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.67 + 2.90i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.10 + 5.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.52 + 5.50i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.308 + 0.178i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.07iT - 71T^{2} \)
73 \( 1 + (3.41 - 5.91i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.52 + 7.83i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.809iT - 83T^{2} \)
89 \( 1 + (-2.00 - 3.47i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.87T + 97T^{2} \)
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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

−10.79405748395458755941879466117, −9.901440532904254735587495130343, −8.848808192435621678939021916632, −8.382178525716172996920162624209, −6.90112083608417046164054417213, −6.46735143284273229443214309602, −5.37383728567627363487110357318, −4.63256885407861379225253583504, −3.09171479981977406632086455462, −1.94633555190176446168461615555, 1.13722841230434069838106195012, 2.25235539886854470230046718117, 3.84234052344161131484426654714, 4.60026135866026857381909590783, 5.75070613950152578605051953506, 6.50850770983103542021359724180, 7.88287532777699557818778415608, 8.731486570706896007191632808358, 9.847682051497612710058304772787, 10.19394889535844404551166295836

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