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Properties

Label 2-630-21.20-c1-0-5
Degree $2$
Conductor $630$
Sign $-0.405 - 0.914i$
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 − 5-s + (2.27 + 1.35i)7-s − i·8-s − i·10-s − 2.71i·11-s + 6.54i·13-s + (−1.35 + 2.27i)14-s + 16-s + 1.53·17-s + 2.30i·19-s + 20-s + 2.71·22-s + 3.83i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.447·5-s + (0.858 + 0.512i)7-s − 0.353i·8-s − 0.316i·10-s − 0.817i·11-s + 1.81i·13-s + (−0.362 + 0.607i)14-s + 0.250·16-s + 0.371·17-s + 0.528i·19-s + 0.223·20-s + 0.577·22-s + 0.799i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.405 - 0.914i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.405 - 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.701933 + 1.07921i\)
\(L(\frac12)\) \(\approx\) \(0.701933 + 1.07921i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + (-2.27 - 1.35i)T \)
good11 \( 1 + 2.71iT - 11T^{2} \)
13 \( 1 - 6.54iT - 13T^{2} \)
17 \( 1 - 1.53T + 17T^{2} \)
19 \( 1 - 2.30iT - 19T^{2} \)
23 \( 1 - 3.83iT - 23T^{2} \)
29 \( 1 - 3.83iT - 29T^{2} \)
31 \( 1 - 3.25iT - 31T^{2} \)
37 \( 1 - 3.01T + 37T^{2} \)
41 \( 1 + 6.54T + 41T^{2} \)
43 \( 1 + 0.468T + 43T^{2} \)
47 \( 1 + 9.11T + 47T^{2} \)
53 \( 1 - 9.25iT - 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + 4.78iT - 61T^{2} \)
67 \( 1 - 13.5T + 67T^{2} \)
71 \( 1 + 2.30iT - 71T^{2} \)
73 \( 1 + 11.4iT - 73T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 + 9.60T + 89T^{2} \)
97 \( 1 + 16.9iT - 97T^{2} \)
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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

−11.07770482230512177042955428909, −9.781072303649840579586037177728, −8.852151538881282727978788024135, −8.297914521510437243603775997551, −7.35532577827213637205414944375, −6.44331861901830511910011182899, −5.42711454321141432510529912913, −4.53129085566479997725452248163, −3.43736807560252439101252328985, −1.63581865483936856832754839071, 0.76351976493439565249406061412, 2.35277964459414442982569383880, 3.59926425775898910976077612102, 4.63490750921561502154752564075, 5.42085874457573624346914823522, 6.93161839183446099024335482725, 7.965136810228110991987107289249, 8.369722098639694319081530251091, 9.819844403056316235478226230498, 10.31582593687276613471572585321

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