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Properties

Label 2-630-63.5-c1-0-7
Degree $2$
Conductor $630$
Sign $0.778 - 0.627i$
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 + (1.51 + 0.840i)3-s − 4-s + (0.5 + 0.866i)5-s + (0.840 − 1.51i)6-s + (−1.15 + 2.38i)7-s + i·8-s + (1.58 + 2.54i)9-s + (0.866 − 0.5i)10-s + (−1.81 − 1.04i)11-s + (−1.51 − 0.840i)12-s + (0.413 + 0.238i)13-s + (2.38 + 1.15i)14-s + (0.0292 + 1.73i)15-s + 16-s + (1.44 + 2.49i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.874 + 0.485i)3-s − 0.5·4-s + (0.223 + 0.387i)5-s + (0.343 − 0.618i)6-s + (−0.434 + 0.900i)7-s + 0.353i·8-s + (0.528 + 0.848i)9-s + (0.273 − 0.158i)10-s + (−0.546 − 0.315i)11-s + (−0.437 − 0.242i)12-s + (0.114 + 0.0662i)13-s + (0.636 + 0.307i)14-s + (0.00755 + 0.447i)15-s + 0.250·16-s + (0.349 + 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.778 - 0.627i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.778 - 0.627i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68720 + 0.595655i\)
\(L(\frac12)\) \(\approx\) \(1.68720 + 0.595655i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.51 - 0.840i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (1.15 - 2.38i)T \)
good11 \( 1 + (1.81 + 1.04i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.413 - 0.238i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.44 - 2.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.03 - 2.32i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.75 - 1.01i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.74 + 1.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.30iT - 31T^{2} \)
37 \( 1 + (1.93 - 3.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.03 + 6.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.96 + 5.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.105T + 47T^{2} \)
53 \( 1 + (-4.27 + 2.46i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.81T + 59T^{2} \)
61 \( 1 - 4.29iT - 61T^{2} \)
67 \( 1 - 5.47T + 67T^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 + (-5.64 + 3.26i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 + (5.58 + 9.68i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.59 + 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.98 - 4.60i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.30263239487549908260942813009, −10.07237677522386388967933918025, −8.995686764904920986566304710440, −8.418050418073580177206240423993, −7.39180559926070563723070635133, −5.95788764343995725322506491726, −5.07307845101092921152021970230, −3.66389240952283408735980431230, −2.97860878961055757483949218349, −1.91685690041785555534878035825, 0.941302265196472342980017207479, 2.72419405909903895977869734245, 3.91036839261485520532752412021, 4.98390623252152323745485817613, 6.21497544650020950168855157233, 7.17297938128807294407797590684, 7.71114415603738788705062243234, 8.597797040626877085631303522031, 9.634310496328074215362217855691, 9.984800915540305767516286582946

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