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Properties

Label 2-630-9.7-c1-0-20
Degree $2$
Conductor $630$
Sign $-0.600 + 0.799i$
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 + (0.403 − 1.68i)3-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (1.25 + 1.19i)6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (−2.67 − 1.35i)9-s + 0.999·10-s + (1.25 − 2.17i)11-s + (−1.66 + 0.492i)12-s + (−0.757 − 1.31i)13-s + (−0.499 − 0.866i)14-s + (−1.66 + 0.492i)15-s + (−0.5 + 0.866i)16-s − 0.320·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.232 − 0.972i)3-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (0.513 + 0.486i)6-s + (−0.188 + 0.327i)7-s + 0.353·8-s + (−0.891 − 0.452i)9-s + 0.316·10-s + (0.379 − 0.656i)11-s + (−0.479 + 0.142i)12-s + (−0.209 − 0.363i)13-s + (−0.133 − 0.231i)14-s + (−0.428 + 0.127i)15-s + (−0.125 + 0.216i)16-s − 0.0778·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.600 + 0.799i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.600 + 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.332226 - 0.665388i\)
\(L(\frac12)\) \(\approx\) \(0.332226 - 0.665388i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.403 + 1.68i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good11 \( 1 + (-1.25 + 2.17i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.757 + 1.31i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.320T + 17T^{2} \)
19 \( 1 + 2.70T + 19T^{2} \)
23 \( 1 + (3.07 + 5.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.563 + 0.976i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.193 + 0.334i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.86T + 37T^{2} \)
41 \( 1 + (5.38 + 9.32i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.01 - 8.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.320 + 0.555i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.83T + 53T^{2} \)
59 \( 1 + (6.16 + 10.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0966 + 0.167i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.546 - 0.947i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 - 5.64T + 73T^{2} \)
79 \( 1 + (-2.32 + 4.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.83 + 4.91i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 16.0T + 89T^{2} \)
97 \( 1 + (-1.30 + 2.26i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.14726199922990404607096612172, −8.992585109665340280157353689666, −8.478223980424968418797469007285, −7.76594210071245531427296637225, −6.67525541078825684746848295364, −6.09027947830869915094130124062, −4.98909457763434922528741584316, −3.50633408803594850530395842540, −2.03669592322015308852126482144, −0.42763240224140011270624793208, 2.02276088175864309062206020874, 3.35327418814461569589834728172, 4.09595853405597501451085495198, 5.09735108856650814189342592088, 6.53476794460478238634470414641, 7.56663525481023926380374692264, 8.539684177692311848261985350382, 9.399203355424154388516739650968, 10.07089675212795181394859949775, 10.69347050642203712582631510889

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