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Properties

Label 2-630-63.59-c1-0-20
Degree $2$
Conductor $630$
Sign $0.815 + 0.578i$
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  + (−0.866 − 0.5i)2-s + (1.61 − 0.619i)3-s + (0.499 + 0.866i)4-s + 5-s + (−1.71 − 0.271i)6-s + (−2.63 − 0.211i)7-s − 0.999i·8-s + (2.23 − 2.00i)9-s + (−0.866 − 0.5i)10-s + 4.91i·11-s + (1.34 + 1.09i)12-s + (3.91 + 2.26i)13-s + (2.17 + 1.50i)14-s + (1.61 − 0.619i)15-s + (−0.5 + 0.866i)16-s + (2.06 − 3.57i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.933 − 0.357i)3-s + (0.249 + 0.433i)4-s + 0.447·5-s + (−0.698 − 0.110i)6-s + (−0.996 − 0.0800i)7-s − 0.353i·8-s + (0.743 − 0.668i)9-s + (−0.273 − 0.158i)10-s + 1.48i·11-s + (0.388 + 0.314i)12-s + (1.08 + 0.627i)13-s + (0.582 + 0.401i)14-s + (0.417 − 0.160i)15-s + (−0.125 + 0.216i)16-s + (0.501 − 0.868i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.815 + 0.578i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.815 + 0.578i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54546 - 0.492904i\)
\(L(\frac12)\) \(\approx\) \(1.54546 - 0.492904i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-1.61 + 0.619i)T \)
5 \( 1 - T \)
7 \( 1 + (2.63 + 0.211i)T \)
good11 \( 1 - 4.91iT - 11T^{2} \)
13 \( 1 + (-3.91 - 2.26i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.06 + 3.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.89 + 3.98i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.03iT - 23T^{2} \)
29 \( 1 + (2.46 - 1.42i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.36 + 0.789i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.05 - 8.76i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.31 - 7.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.88 + 4.99i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.227 + 0.393i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.7 + 6.21i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.48 + 9.49i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.66 - 2.11i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.96 + 3.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.14iT - 71T^{2} \)
73 \( 1 + (-2.06 - 1.19i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.86 - 8.42i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.42 + 2.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.95 - 5.11i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.74 + 1.00i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.954461718244050182366211821619, −9.700449017211204028573342225078, −9.021809908281755777578806004069, −7.967306144392845926669063987900, −6.97972482747020416806092811648, −6.53109412469823886387503553793, −4.77124597117570021479088797815, −3.43676234442359465379040876449, −2.57765738436338329189751777620, −1.27949262916204776230724586535, 1.33970569331262087589775224058, 3.11484636653292605149036007013, 3.63650800064705876521552896204, 5.67726147214177090724114618388, 5.98963512802217445602179002270, 7.45214545869196791432050989918, 8.164714839434925199914831255137, 9.001576990784159467595811591577, 9.633624321453795527733203811880, 10.39331349808264550976023770200

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