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Properties

Label 2-630-7.4-c1-0-10
Degree $2$
Conductor $630$
Sign $-0.895 + 0.444i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 2.64·7-s + 0.999·8-s + (0.499 − 0.866i)10-s + (0.322 − 0.559i)11-s − 4.64·13-s + (1.32 + 2.29i)14-s + (−0.5 − 0.866i)16-s + (3.64 − 6.31i)17-s + (−3.14 − 5.44i)19-s − 0.999·20-s − 0.645·22-s + (−1.5 − 2.59i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s − 0.999·7-s + 0.353·8-s + (0.158 − 0.273i)10-s + (0.0973 − 0.168i)11-s − 1.28·13-s + (0.353 + 0.612i)14-s + (−0.125 − 0.216i)16-s + (0.884 − 1.53i)17-s + (−0.721 − 1.24i)19-s − 0.223·20-s − 0.137·22-s + (−0.312 − 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.123467 - 0.527049i\)
\(L(\frac12)\) \(\approx\) \(0.123467 - 0.527049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + 2.64T \)
good11 \( 1 + (-0.322 + 0.559i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.64T + 13T^{2} \)
17 \( 1 + (-3.64 + 6.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.14 + 5.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.96 + 8.60i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.64T + 41T^{2} \)
43 \( 1 - 0.708T + 43T^{2} \)
47 \( 1 + (5.79 + 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.14 - 8.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.645 - 1.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.354 + 0.613i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.64 - 13.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.354 + 0.613i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.70T + 83T^{2} \)
89 \( 1 + (-7.29 - 12.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−10.14626086030412772057862312502, −9.495558448097628949449819636971, −8.833357367315313743326572519964, −7.40341770889644702935745542057, −6.92832371227327734065589789825, −5.63207655404382627329331334129, −4.48380589065957832534611729505, −3.11840617142882587567613849143, −2.42038277257109046694171738496, −0.31838991124440795983823507609, 1.73411149581772625334542665970, 3.42300626933364224939269565460, 4.61616311140100284708551210996, 5.82978080314538939436787727242, 6.36567479067815922946212167455, 7.56319049433286618083646253095, 8.225972892411194112427010108782, 9.338034029400955565684528837042, 9.947518073045839123578676090706, 10.52717641242615059626564275254

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