Properties

Label 2-630-5.4-c3-0-4
Degree $2$
Conductor $630$
Sign $-0.894 + 0.447i$
Analytic cond. $37.1712$
Root an. cond. $6.09681$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 4·4-s + (−10 + 5i)5-s − 7i·7-s − 8i·8-s + (−10 − 20i)10-s + 37·11-s + 51i·13-s + 14·14-s + 16·16-s + 41i·17-s + 108·19-s + (40 − 20i)20-s + 74i·22-s + 70i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.894 + 0.447i)5-s − 0.377i·7-s − 0.353i·8-s + (−0.316 − 0.632i)10-s + 1.01·11-s + 1.08i·13-s + 0.267·14-s + 0.250·16-s + 0.584i·17-s + 1.30·19-s + (0.447 − 0.223i)20-s + 0.717i·22-s + 0.634i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(37.1712\)
Root analytic conductor: \(6.09681\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :3/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6372723578\)
\(L(\frac12)\) \(\approx\) \(0.6372723578\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
3 \( 1 \)
5 \( 1 + (10 - 5i)T \)
7 \( 1 + 7iT \)
good11 \( 1 - 37T + 1.33e3T^{2} \)
13 \( 1 - 51iT - 2.19e3T^{2} \)
17 \( 1 - 41iT - 4.91e3T^{2} \)
19 \( 1 - 108T + 6.85e3T^{2} \)
23 \( 1 - 70iT - 1.21e4T^{2} \)
29 \( 1 + 249T + 2.43e4T^{2} \)
31 \( 1 + 134T + 2.97e4T^{2} \)
37 \( 1 - 334iT - 5.06e4T^{2} \)
41 \( 1 + 206T + 6.89e4T^{2} \)
43 \( 1 + 376iT - 7.95e4T^{2} \)
47 \( 1 + 287iT - 1.03e5T^{2} \)
53 \( 1 - 6iT - 1.48e5T^{2} \)
59 \( 1 + 2T + 2.05e5T^{2} \)
61 \( 1 + 940T + 2.26e5T^{2} \)
67 \( 1 + 106iT - 3.00e5T^{2} \)
71 \( 1 + 456T + 3.57e5T^{2} \)
73 \( 1 - 650iT - 3.89e5T^{2} \)
79 \( 1 - 1.23e3T + 4.93e5T^{2} \)
83 \( 1 + 428iT - 5.71e5T^{2} \)
89 \( 1 + 220T + 7.04e5T^{2} \)
97 \( 1 - 1.05e3iT - 9.12e5T^{2} \)
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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

−10.72603460311465949298421503942, −9.597723325646776451877683842247, −8.915062921087237537471510431304, −7.84106422367758109736292742404, −7.13224424003587041105373076634, −6.49550488558850447515671190071, −5.26258834379318647545066213802, −4.04885445611249140480425287445, −3.49510918714128743895266790079, −1.48231077625781851414051174005, 0.20234933933853706015688733059, 1.38473405194428741521567792849, 2.99028475327962788497942208885, 3.81825252562997186121760940613, 4.90875091422231700969318369327, 5.81325642214351136113544996696, 7.29214796333505267818962512698, 7.989100203037682398572975020342, 9.103949994504107886569059104777, 9.489407470476623909613195808974

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