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Properties

Label 2-630-35.4-c1-0-12
Degree $2$
Conductor $630$
Sign $0.308 + 0.951i$
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 + (0.499 − 0.866i)4-s + (−1.52 + 1.63i)5-s + (1.13 − 2.38i)7-s − 0.999i·8-s + (−0.499 + 2.17i)10-s + (0.866 − 1.5i)11-s − 6.50i·13-s + (−0.209 − 2.63i)14-s + (−0.5 − 0.866i)16-s + (1.73 + i)17-s + (1.63 + 2.83i)19-s + (0.656 + 2.13i)20-s − 1.73i·22-s + (2.83 − 1.63i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.680 + 0.732i)5-s + (0.429 − 0.902i)7-s − 0.353i·8-s + (−0.158 + 0.689i)10-s + (0.261 − 0.452i)11-s − 1.80i·13-s + (−0.0559 − 0.704i)14-s + (−0.125 − 0.216i)16-s + (0.420 + 0.242i)17-s + (0.375 + 0.650i)19-s + (0.146 + 0.477i)20-s − 0.369i·22-s + (0.591 − 0.341i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.308 + 0.951i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.308 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56644 - 1.13839i\)
\(L(\frac12)\) \(\approx\) \(1.56644 - 1.13839i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (1.52 - 1.63i)T \)
7 \( 1 + (-1.13 + 2.38i)T \)
good11 \( 1 + (-0.866 + 1.5i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.50iT - 13T^{2} \)
17 \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.63 - 2.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.83 + 1.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.24T + 29T^{2} \)
31 \( 1 + (0.137 - 0.238i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.362 - 0.209i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.20T + 41T^{2} \)
43 \( 1 + 6.09iT - 43T^{2} \)
47 \( 1 + (8.03 - 4.63i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.53 + 3.77i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.85 - 10.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.27 - 9.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.54 - 4.35i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + (4.54 + 2.62i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.13 - 3.70i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.27iT - 83T^{2} \)
89 \( 1 + (-4.77 - 8.27i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.8iT - 97T^{2} \)
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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

−10.44049605972445264404573296015, −10.14433988534802961857872743034, −8.410932887747564369995750089624, −7.75284957208196275160353324331, −6.84200275163671299338081170018, −5.79961332928262825389567516599, −4.70771640125856704149385777738, −3.60473589023565443272435915217, −2.94135932817316883878685630751, −0.965333450820103218499543335589, 1.73550558481644195893838762308, 3.24939882714938864378452807053, 4.64500601301302180878621889373, 4.89246625996841000451974435859, 6.28224846056581781623524082786, 7.14376739699208823461553326756, 8.133863588175962922775993180400, 8.942168144007361492624076126198, 9.598937675708734223830474168466, 11.28519841270200322039660404632

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