Properties

Label 2-630-5.2-c2-0-14
Degree $2$
Conductor $630$
Sign $0.999 - 0.0260i$
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  + (−1 − i)2-s + 2i·4-s + (4.32 + 2.51i)5-s + (−1.87 − 1.87i)7-s + (2 − 2i)8-s + (−1.80 − 6.83i)10-s + 14.0·11-s + (−6.03 + 6.03i)13-s + 3.74i·14-s − 4·16-s + (9.54 + 9.54i)17-s − 21.6i·19-s + (−5.03 + 8.64i)20-s + (−14.0 − 14.0i)22-s + (−0.423 + 0.423i)23-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + 0.5i·4-s + (0.864 + 0.503i)5-s + (−0.267 − 0.267i)7-s + (0.250 − 0.250i)8-s + (−0.180 − 0.683i)10-s + 1.28·11-s + (−0.464 + 0.464i)13-s + 0.267i·14-s − 0.250·16-s + (0.561 + 0.561i)17-s − 1.14i·19-s + (−0.251 + 0.432i)20-s + (−0.640 − 0.640i)22-s + (−0.0184 + 0.0184i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0260i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.999 - 0.0260i$
Analytic conductor: \(17.1662\)
Root analytic conductor: \(4.14321\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1),\ 0.999 - 0.0260i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.681599227\)
\(L(\frac12)\) \(\approx\) \(1.681599227\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 \)
5 \( 1 + (-4.32 - 2.51i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good11 \( 1 - 14.0T + 121T^{2} \)
13 \( 1 + (6.03 - 6.03i)T - 169iT^{2} \)
17 \( 1 + (-9.54 - 9.54i)T + 289iT^{2} \)
19 \( 1 + 21.6iT - 361T^{2} \)
23 \( 1 + (0.423 - 0.423i)T - 529iT^{2} \)
29 \( 1 + 11.2iT - 841T^{2} \)
31 \( 1 + 16.4T + 961T^{2} \)
37 \( 1 + (-47.6 - 47.6i)T + 1.36e3iT^{2} \)
41 \( 1 - 44.0T + 1.68e3T^{2} \)
43 \( 1 + (46.7 - 46.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-20.3 - 20.3i)T + 2.20e3iT^{2} \)
53 \( 1 + (-18.6 + 18.6i)T - 2.80e3iT^{2} \)
59 \( 1 - 13.4iT - 3.48e3T^{2} \)
61 \( 1 + 10.8T + 3.72e3T^{2} \)
67 \( 1 + (-72.2 - 72.2i)T + 4.48e3iT^{2} \)
71 \( 1 - 64.1T + 5.04e3T^{2} \)
73 \( 1 + (-51.4 + 51.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 157. iT - 6.24e3T^{2} \)
83 \( 1 + (-76.9 + 76.9i)T - 6.88e3iT^{2} \)
89 \( 1 - 37.6iT - 7.92e3T^{2} \)
97 \( 1 + (-97.2 - 97.2i)T + 9.40e3iT^{2} \)
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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

−10.26459997645954696660403642476, −9.528900263095942101815564327260, −9.059182572340855018790657100911, −7.77967821658943099267451568829, −6.77373553338502997581121391774, −6.16303365330612095577681709901, −4.68584986631829450238773574718, −3.51335375361311705289427807816, −2.38077049649982409121390212105, −1.15301489036159367260902543831, 0.909905903893603426238359810735, 2.19503388325303866782985629923, 3.81735677378130550229696041701, 5.21199760301494343424651319834, 5.88849313217967073244231014356, 6.77621348163658503729022477595, 7.79842848609960721616613265478, 8.796138184446271616263944961553, 9.500307091343171842524511709641, 9.989898173076334789386169524538

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