Properties

Label 2-630-5.4-c1-0-10
Degree $2$
Conductor $630$
Sign $-0.447 + 0.894i$
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  i·2-s − 4-s + (−1 + 2i)5-s i·7-s + i·8-s + (2 + i)10-s − 2·11-s − 6i·13-s − 14-s + 16-s − 4i·17-s + 6·19-s + (1 − 2i)20-s + 2i·22-s − 8i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.447 + 0.894i)5-s − 0.377i·7-s + 0.353i·8-s + (0.632 + 0.316i)10-s − 0.603·11-s − 1.66i·13-s − 0.267·14-s + 0.250·16-s − 0.970i·17-s + 1.37·19-s + (0.223 − 0.447i)20-s + 0.426i·22-s − 1.66i·23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.540391 - 0.874372i\)
\(L(\frac12)\) \(\approx\) \(0.540391 - 0.874372i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + (1 - 2i)T \)
7 \( 1 + iT \)
good11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−10.37876365930032148455785244878, −9.881648777276883356619630732910, −8.498726438717378099000866763611, −7.73686308934891938485147179089, −6.91389498751743686640947926460, −5.59335365391892359207198904602, −4.60379425873459331820356066447, −3.23315800775245169447093229103, −2.71974602446305936445016244280, −0.59475165323952871774110086794, 1.56109742130445673011583446443, 3.49749682254059423203596940462, 4.58926868205107168002095347823, 5.37835025956207649602262342240, 6.36832411843410556343132192733, 7.52358689039023621805349377382, 8.109147148284162153904266411237, 9.210644828347075092309504412288, 9.508812104362100193697388887467, 10.96147662229947354013546302742

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