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Properties

Label 2-630-315.209-c1-0-7
Degree $2$
Conductor $630$
Sign $-0.991 - 0.126i$
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 + (1.42 + 0.989i)3-s + (−0.499 − 0.866i)4-s + (−1.07 + 1.96i)5-s + (−1.56 + 0.736i)6-s + (−2.59 + 0.507i)7-s + 0.999·8-s + (1.04 + 2.81i)9-s + (−1.16 − 1.90i)10-s + (4.30 + 2.48i)11-s + (0.146 − 1.72i)12-s + (−1.15 − 1.99i)13-s + (0.858 − 2.50i)14-s + (−3.46 + 1.72i)15-s + (−0.5 + 0.866i)16-s + 4.05i·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.820 + 0.571i)3-s + (−0.249 − 0.433i)4-s + (−0.479 + 0.877i)5-s + (−0.640 + 0.300i)6-s + (−0.981 + 0.191i)7-s + 0.353·8-s + (0.347 + 0.937i)9-s + (−0.367 − 0.603i)10-s + (1.29 + 0.749i)11-s + (0.0421 − 0.498i)12-s + (−0.319 − 0.553i)13-s + (0.229 − 0.668i)14-s + (−0.894 + 0.446i)15-s + (−0.125 + 0.216i)16-s + 0.984i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0699497 + 1.09976i\)
\(L(\frac12)\) \(\approx\) \(0.0699497 + 1.09976i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 - 0.989i)T \)
5 \( 1 + (1.07 - 1.96i)T \)
7 \( 1 + (2.59 - 0.507i)T \)
good11 \( 1 + (-4.30 - 2.48i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.15 + 1.99i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.05iT - 17T^{2} \)
19 \( 1 - 1.63iT - 19T^{2} \)
23 \( 1 + (3.70 + 6.41i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.86 + 3.96i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.00 - 3.46i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.4iT - 37T^{2} \)
41 \( 1 + (2.40 + 4.17i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.31 + 1.33i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-9.03 - 5.21i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 + (-5.82 - 10.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.89 - 2.24i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.266 - 0.153i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.38iT - 71T^{2} \)
73 \( 1 + 3.95T + 73T^{2} \)
79 \( 1 + (1.42 - 2.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.60 - 5.54i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.77T + 89T^{2} \)
97 \( 1 + (-4.81 + 8.34i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.40182744088921841676632659353, −10.19394980826310012863814156509, −9.209585123870470291515514019280, −8.457603771969475046638240180045, −7.47220311286390923381509300687, −6.75445357421839921599283453216, −5.82095316818305136461617781918, −4.21752183993205903771212680867, −3.58837331298998245244101896486, −2.22529518243229007036164027656, 0.61062406538289418393549022743, 1.98003760754161394248546727423, 3.55157609277894921829637170063, 3.91954848268307888433617588297, 5.63995070418927190576093651131, 7.00244906944172230980169769635, 7.54224082682806391205768598029, 8.844105577927171121195002863713, 9.170786179953437133181357105608, 9.754404786214109595671499811365

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