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Properties

Label 2-630-9.7-c1-0-10
Degree $2$
Conductor $630$
Sign $0.939 - 0.342i$
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 + (1.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (1.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s − 0.999·8-s + (1.5 + 2.59i)9-s + 0.999·10-s + (−2.5 + 4.33i)11-s − 1.73i·12-s + (1 + 1.73i)13-s + (0.499 + 0.866i)14-s + 1.73i·15-s + (−0.5 + 0.866i)16-s + 7·17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.866 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s + (0.612 − 0.353i)6-s + (−0.188 + 0.327i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + 0.316·10-s + (−0.753 + 1.30i)11-s − 0.499i·12-s + (0.277 + 0.480i)13-s + (0.133 + 0.231i)14-s + 0.447i·15-s + (−0.125 + 0.216i)16-s + 1.69·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28939 + 0.403682i\)
\(L(\frac12)\) \(\approx\) \(2.28939 + 0.403682i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 7T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.5 + 7.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (2.5 + 4.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + (-8.5 + 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.36481483731731213760200487268, −9.877329586696150799536001458915, −9.283660028366166978370695421237, −8.027782512021516604278218624139, −7.30887634933891437708677263848, −5.87995041447642135655848557965, −4.89962583466612061323515071571, −3.87268833340971304738576442859, −2.84021985488528418213806348790, −1.93606746847455394209298259884, 1.16754950605903517617996547617, 3.11533583907217185785944165839, 3.61682435706520419553503276334, 5.37069511477277290592779588442, 5.86058970164482261409402362482, 7.25422731767549217696299840669, 7.83096649882231274882517647183, 8.545688766194194327727523938423, 9.497101907277779084747248568984, 10.32023709047851670443531844759

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