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Properties

Label 2-630-315.194-c1-0-27
Degree $2$
Conductor $630$
Sign $0.863 - 0.504i$
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 + (1.69 + 0.335i)3-s + 4-s + (0.325 + 2.21i)5-s + (1.69 + 0.335i)6-s + (−1.75 − 1.98i)7-s + 8-s + (2.77 + 1.14i)9-s + (0.325 + 2.21i)10-s + (0.702 + 0.405i)11-s + (1.69 + 0.335i)12-s + (0.0126 − 0.0219i)13-s + (−1.75 − 1.98i)14-s + (−0.189 + 3.86i)15-s + 16-s + (2.78 − 1.60i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.981 + 0.193i)3-s + 0.5·4-s + (0.145 + 0.989i)5-s + (0.693 + 0.137i)6-s + (−0.663 − 0.748i)7-s + 0.353·8-s + (0.924 + 0.380i)9-s + (0.102 + 0.699i)10-s + (0.211 + 0.122i)11-s + (0.490 + 0.0969i)12-s + (0.00351 − 0.00609i)13-s + (−0.468 − 0.529i)14-s + (−0.0489 + 0.998i)15-s + 0.250·16-s + (0.675 − 0.390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.863 - 0.504i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.863 - 0.504i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.96689 + 0.803149i\)
\(L(\frac12)\) \(\approx\) \(2.96689 + 0.803149i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 - 0.335i)T \)
5 \( 1 + (-0.325 - 2.21i)T \)
7 \( 1 + (1.75 + 1.98i)T \)
good11 \( 1 + (-0.702 - 0.405i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0126 + 0.0219i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.78 + 1.60i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.22 - 0.705i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.76 - 3.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.38 - 2.52i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.34iT - 31T^{2} \)
37 \( 1 + (9.76 + 5.63i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.973 - 1.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.76 - 1.59i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.51iT - 47T^{2} \)
53 \( 1 + (5.33 + 9.24i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 3.60T + 59T^{2} \)
61 \( 1 - 0.608iT - 61T^{2} \)
67 \( 1 - 1.01iT - 67T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + (6.25 + 10.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 1.69T + 79T^{2} \)
83 \( 1 + (-14.5 + 8.40i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.31 - 4.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.19 - 5.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.52271658762810692709512086134, −9.936916186812033034192210851187, −9.108961152973893960722596466337, −7.63420105540619664394589587384, −7.23599681935661560790009785039, −6.29004604237229000846906252682, −5.01297624236331507889145403223, −3.54612987285719134353102417352, −3.38894693766963578045558393292, −1.93968390712281401641181851930, 1.54102768650211248331169363087, 2.83367749537330364377871074416, 3.78794625862487432024204378474, 4.92965920322856371939356871890, 5.90907852637484912713004564634, 6.88785313482085231799326347943, 8.011309102694142120138940201440, 8.792589603840870941650635459576, 9.472483913875061054515792630224, 10.36264726053231585535642522547

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