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Properties

Label 2-630-35.4-c1-0-5
Degree $2$
Conductor $630$
Sign $0.758 - 0.652i$
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 + (0.133 + 2.23i)5-s + (1.73 + 2i)7-s − 0.999i·8-s + (1.23 + 1.86i)10-s + (−2.5 + 4.33i)11-s − i·13-s + (2.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−1.73 − i)17-s + (3.5 + 6.06i)19-s + (1.99 + 0.999i)20-s + 5i·22-s + (2.59 − 1.5i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.0599 + 0.998i)5-s + (0.654 + 0.755i)7-s − 0.353i·8-s + (0.389 + 0.590i)10-s + (−0.753 + 1.30i)11-s − 0.277i·13-s + (0.668 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.420 − 0.242i)17-s + (0.802 + 1.39i)19-s + (0.447 + 0.223i)20-s + 1.06i·22-s + (0.541 − 0.312i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.758 - 0.652i$
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: \((2,\ 630,\ (\ :1/2),\ 0.758 - 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98282 + 0.735381i\)
\(L(\frac12)\) \(\approx\) \(1.98282 + 0.735381i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.133 - 2.23i)T \)
7 \( 1 + (-1.73 - 2i)T \)
good11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.33 - 2.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 + (-11.2 + 6.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.19 + 3i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-3.46 - 2i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10iT - 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−10.60299864276935092430231333840, −10.19185850458471037543824432530, −9.126271494975588249679236401015, −7.81034364644191906995962133823, −7.18482053576496075458945873756, −5.96293097448725792513442068911, −5.21273478152558914607338496570, −4.10436217818334135475412777466, −2.78439994598385819723604657171, −1.99095214171693134601952006605, 1.01673583712833570281098253119, 2.83540474322920179612059912025, 4.16047965548791650425947388157, 4.97508830335684216912364954600, 5.71449571147239008495961339698, 6.94650119229071343143501555493, 7.86213349096820001720376960733, 8.591632520214981573039576570953, 9.436111713613317334012192752242, 10.90007922742179005637330109847

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