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Properties

Label 2-630-105.2-c1-0-3
Degree $2$
Conductor $630$
Sign $-0.995 + 0.0926i$
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−1.21 + 1.87i)5-s + (0.576 + 2.58i)7-s + (−0.707 − 0.707i)8-s + (−2.12 − 0.686i)10-s + (−1.41 + 0.819i)11-s + (1 − i)13-s + (−2.34 + 1.22i)14-s + (0.500 − 0.866i)16-s + (−2.23 − 0.599i)17-s + (0.274 + 0.158i)19-s + (0.111 − 2.23i)20-s + (−1.15 − 1.15i)22-s + (−8.03 + 2.15i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.542 + 0.839i)5-s + (0.217 + 0.976i)7-s + (−0.249 − 0.249i)8-s + (−0.672 − 0.216i)10-s + (−0.427 + 0.246i)11-s + (0.277 − 0.277i)13-s + (−0.626 + 0.327i)14-s + (0.125 − 0.216i)16-s + (−0.542 − 0.145i)17-s + (0.0629 + 0.0363i)19-s + (0.0250 − 0.499i)20-s + (−0.246 − 0.246i)22-s + (−1.67 + 0.448i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.995 + 0.0926i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.995 + 0.0926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0429069 - 0.923808i\)
\(L(\frac12)\) \(\approx\) \(0.0429069 - 0.923808i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 \)
5 \( 1 + (1.21 - 1.87i)T \)
7 \( 1 + (-0.576 - 2.58i)T \)
good11 \( 1 + (1.41 - 0.819i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 + i)T - 13iT^{2} \)
17 \( 1 + (2.23 + 0.599i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.274 - 0.158i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (8.03 - 2.15i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 1.19T + 29T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.83 - 1.83i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.82iT - 41T^{2} \)
43 \( 1 + (-5.63 + 5.63i)T - 43iT^{2} \)
47 \( 1 + (-1.63 - 6.10i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.01 - 11.2i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.04 + 1.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.158 + 0.274i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.963 - 3.59i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.51iT - 71T^{2} \)
73 \( 1 + (-12.7 - 3.41i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-8.34 - 4.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.0 - 10.0i)T + 83iT^{2} \)
89 \( 1 + (-7.74 + 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.15 + 9.15i)T + 97iT^{2} \)
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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

−11.04579601211366293034393586831, −10.18248068037878349852825685664, −9.132928345611662137977099782639, −8.189744808118535034852160121973, −7.58829584621363009481007813314, −6.51500381130091999123642928258, −5.77026445076895154952366666437, −4.67911961584157027901189762689, −3.51604535573328213237013121240, −2.34175853741531241804832487870, 0.46581695605723657178385296266, 1.95048050698717068642828056435, 3.64780863326547371541027080302, 4.30916700146651371531883461281, 5.25247319415287159684768082774, 6.50553387799241937713561282608, 7.77597168311627908608247025306, 8.384952728725721198869495879500, 9.374837658425653020022400154605, 10.32906573761971418420853841161

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