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Properties

Label 2-630-35.27-c1-0-4
Degree $2$
Conductor $630$
Sign $-0.865 - 0.501i$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.158 + 2.23i)5-s + (−2.47 − 0.941i)7-s + (−0.707 + 0.707i)8-s + (−1.46 + 1.68i)10-s − 2.82·11-s + (4.23 + 4.23i)13-s + (−1.08 − 2.41i)14-s − 1.00·16-s + (−3.69 + 3.69i)17-s − 1.39·19-s + (−2.23 + 0.158i)20-s + (−2.00 − 2.00i)22-s + (−0.414 + 0.414i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.0708 + 0.997i)5-s + (−0.934 − 0.355i)7-s + (−0.250 + 0.250i)8-s + (−0.463 + 0.534i)10-s − 0.852·11-s + (1.17 + 1.17i)13-s + (−0.289 − 0.645i)14-s − 0.250·16-s + (−0.896 + 0.896i)17-s − 0.321·19-s + (−0.498 + 0.0354i)20-s + (−0.426 − 0.426i)22-s + (−0.0863 + 0.0863i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.865 - 0.501i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.865 - 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.345494 + 1.28631i\)
\(L(\frac12)\) \(\approx\) \(0.345494 + 1.28631i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (-0.158 - 2.23i)T \)
7 \( 1 + (2.47 + 0.941i)T \)
good11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 + (-4.23 - 4.23i)T + 13iT^{2} \)
17 \( 1 + (3.69 - 3.69i)T - 17iT^{2} \)
19 \( 1 + 1.39T + 19T^{2} \)
23 \( 1 + (0.414 - 0.414i)T - 23iT^{2} \)
29 \( 1 - 0.828iT - 29T^{2} \)
31 \( 1 + 1.53iT - 31T^{2} \)
37 \( 1 + (-2.58 - 2.58i)T + 37iT^{2} \)
41 \( 1 - 3.69iT - 41T^{2} \)
43 \( 1 + (-4 + 4i)T - 43iT^{2} \)
47 \( 1 + (-1.08 + 1.08i)T - 47iT^{2} \)
53 \( 1 + (-8.24 + 8.24i)T - 53iT^{2} \)
59 \( 1 + 9.23T + 59T^{2} \)
61 \( 1 - 6.43iT - 61T^{2} \)
67 \( 1 + (-10.4 - 10.4i)T + 67iT^{2} \)
71 \( 1 - 0.585T + 71T^{2} \)
73 \( 1 + (4.14 + 4.14i)T + 73iT^{2} \)
79 \( 1 - 5.07iT - 79T^{2} \)
83 \( 1 + (-5.31 - 5.31i)T + 83iT^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + (-4.59 + 4.59i)T - 97iT^{2} \)
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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

−10.91183366436586592193495591551, −10.25022738737875949426528174187, −9.155109492262162926762759279640, −8.206999106036984476821497151304, −7.10345545359887299538368294879, −6.48282373581259059879686464835, −5.83869117838671226006658359118, −4.27045996579365237105478331308, −3.52270719826472511609890152611, −2.29691410398423358224731106072, 0.59415950815536196049982958232, 2.38282608875464581931814794341, 3.46404436377540382081902565716, 4.64535777917241583771524069681, 5.58891965643535954393360631435, 6.27202445863378307005592449176, 7.68226934564017030283846026555, 8.720345021828329130934484930333, 9.358841768592333675025758746529, 10.35999363213274111087614648796

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