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Properties

Label 2-630-7.5-c2-0-3
Degree $2$
Conductor $630$
Sign $-0.0172 - 0.999i$
Analytic cond. $17.1662$
Root an. cond. $4.14321$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (1.93 + 1.11i)5-s + (−2.59 + 6.50i)7-s − 2.82·8-s + (2.73 − 1.58i)10-s + (5.13 + 8.89i)11-s − 7.02i·13-s + (6.12 + 7.77i)14-s + (−2.00 + 3.46i)16-s + (−27.4 + 15.8i)17-s + (−26.9 − 15.5i)19-s − 4.47i·20-s + 14.5·22-s + (−11.8 + 20.5i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + (−0.370 + 0.928i)7-s − 0.353·8-s + (0.273 − 0.158i)10-s + (0.466 + 0.808i)11-s − 0.540i·13-s + (0.437 + 0.555i)14-s + (−0.125 + 0.216i)16-s + (−1.61 + 0.933i)17-s + (−1.41 − 0.818i)19-s − 0.223i·20-s + 0.660·22-s + (−0.514 + 0.891i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0172 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0172 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0172 - 0.999i$
Analytic conductor: \(17.1662\)
Root analytic conductor: \(4.14321\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1),\ -0.0172 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.102528636\)
\(L(\frac12)\) \(\approx\) \(1.102528636\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 \)
5 \( 1 + (-1.93 - 1.11i)T \)
7 \( 1 + (2.59 - 6.50i)T \)
good11 \( 1 + (-5.13 - 8.89i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 7.02iT - 169T^{2} \)
17 \( 1 + (27.4 - 15.8i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (26.9 + 15.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (11.8 - 20.5i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 9.19T + 841T^{2} \)
31 \( 1 + (-17.4 + 10.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (24.0 - 41.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 65.1iT - 1.68e3T^{2} \)
43 \( 1 + 3.03T + 1.84e3T^{2} \)
47 \( 1 + (-53.6 - 30.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-0.690 - 1.19i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-95.1 + 54.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (34.3 + 19.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (7.95 + 13.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 53.3T + 5.04e3T^{2} \)
73 \( 1 + (-62.6 + 36.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (53.2 - 92.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 49.4iT - 6.88e3T^{2} \)
89 \( 1 + (-142. - 82.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 49.4iT - 9.40e3T^{2} \)
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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

−10.67874165575758158025885548593, −9.790162902948498563328713187881, −9.076616897425843381744795183573, −8.230431758457581578470142882037, −6.66639824668245659182952465207, −6.17167944664507287348468719478, −4.96793526556183784524584534934, −4.00642537591694223507966732626, −2.65231109517236286067485317558, −1.84437501232897187007020613582, 0.33419174427134459413070788457, 2.24447470898618180944206367821, 3.85412573275727623876582032825, 4.44866436769456770424905305386, 5.78841469308543314456218852730, 6.60210259018363290808216582688, 7.21578226101019333563486755598, 8.603211969941228743265100103619, 8.972130868929332815277177738943, 10.22889508563712574770900382881

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