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Properties

Label 2-630-63.58-c1-0-23
Degree $2$
Conductor $630$
Sign $0.204 + 0.978i$
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  + 2-s + (−1.35 − 1.08i)3-s + 4-s + (0.5 + 0.866i)5-s + (−1.35 − 1.08i)6-s + (−0.710 − 2.54i)7-s + 8-s + (0.653 + 2.92i)9-s + (0.5 + 0.866i)10-s + (−0.560 + 0.971i)11-s + (−1.35 − 1.08i)12-s + (3.48 − 6.03i)13-s + (−0.710 − 2.54i)14-s + (0.262 − 1.71i)15-s + 16-s + (−0.317 − 0.549i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.780 − 0.625i)3-s + 0.5·4-s + (0.223 + 0.387i)5-s + (−0.551 − 0.442i)6-s + (−0.268 − 0.963i)7-s + 0.353·8-s + (0.217 + 0.975i)9-s + (0.158 + 0.273i)10-s + (−0.169 + 0.292i)11-s + (−0.390 − 0.312i)12-s + (0.966 − 1.67i)13-s + (−0.189 − 0.681i)14-s + (0.0677 − 0.442i)15-s + 0.250·16-s + (−0.0769 − 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.204 + 0.978i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.204 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34975 - 1.09706i\)
\(L(\frac12)\) \(\approx\) \(1.34975 - 1.09706i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.35 + 1.08i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.710 + 2.54i)T \)
good11 \( 1 + (0.560 - 0.971i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.48 + 6.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.317 + 0.549i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.57 + 2.72i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.29 + 3.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.243 + 0.421i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.14T + 31T^{2} \)
37 \( 1 + (2.03 - 3.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.96 + 3.41i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.71 + 8.16i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.61T + 47T^{2} \)
53 \( 1 + (-6.28 - 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 9.57T + 59T^{2} \)
61 \( 1 - 7.48T + 61T^{2} \)
67 \( 1 - 6.38T + 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (-4.16 - 7.20i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 + (-6.68 - 11.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.22 - 5.58i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.52 + 9.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.59136390477408755969197876274, −10.05593827309945535226003517204, −8.334903613518814770466209400851, −7.43592997458458610518350578722, −6.72217406445137120023604305884, −5.88724340856311083885057833781, −5.03362301528400723675388802188, −3.80650689465735288458515866267, −2.57893372542462448906829097234, −0.884478094351600216223246375174, 1.73448439773393695831013855363, 3.41499594839722213605594561008, 4.33503557411610305454266838431, 5.37761217299128827094739266814, 6.03769450757511404991982804385, 6.74566688448047877886851405822, 8.304876484008046909525210959462, 9.244538381191844188949184055127, 9.889216605009507340656613291219, 11.10735603033915291411598191962

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