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Properties

Label 2-630-105.23-c1-0-10
Degree $2$
Conductor $630$
Sign $0.918 + 0.395i$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (1.87 + 1.21i)5-s + (−1.94 − 1.78i)7-s + (0.707 − 0.707i)8-s + (2.12 + 0.686i)10-s + (2.64 − 1.52i)11-s + (1 + i)13-s + (−2.34 − 1.22i)14-s + (0.500 − 0.866i)16-s + (1.11 − 4.16i)17-s + (5.47 + 3.15i)19-s + (2.23 + 0.111i)20-s + (2.15 − 2.15i)22-s + (−0.435 − 1.62i)23-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.433 − 0.249i)4-s + (0.839 + 0.542i)5-s + (−0.736 − 0.676i)7-s + (0.249 − 0.249i)8-s + (0.672 + 0.216i)10-s + (0.797 − 0.460i)11-s + (0.277 + 0.277i)13-s + (−0.626 − 0.327i)14-s + (0.125 − 0.216i)16-s + (0.270 − 1.01i)17-s + (1.25 + 0.724i)19-s + (0.499 + 0.0250i)20-s + (0.460 − 0.460i)22-s + (−0.0908 − 0.339i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.918 + 0.395i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.918 + 0.395i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.45946 - 0.506344i\)
\(L(\frac12)\) \(\approx\) \(2.45946 - 0.506344i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 \)
5 \( 1 + (-1.87 - 1.21i)T \)
7 \( 1 + (1.94 + 1.78i)T \)
good11 \( 1 + (-2.64 + 1.52i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 + (-1.11 + 4.16i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-5.47 - 3.15i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.435 + 1.62i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 5.88T + 29T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.83 - 6.83i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.82iT - 41T^{2} \)
43 \( 1 + (7.63 + 7.63i)T + 43iT^{2} \)
47 \( 1 + (0.305 - 0.0819i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.57 - 0.422i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.99 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.15 - 5.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (14.5 + 3.89i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 1.86iT - 71T^{2} \)
73 \( 1 + (0.982 - 3.66i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.14 - 1.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.67 + 8.67i)T - 83iT^{2} \)
89 \( 1 + (-6.32 + 10.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.84 - 5.84i)T - 97iT^{2} \)
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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

−10.51461261016213402100148433105, −9.809216361691799441800318140133, −9.131346560244985605553034108477, −7.58701625070870559338996250044, −6.72332802658257610014260802118, −6.08572640654303372569998351893, −5.06553712373560895040837650752, −3.68342273025410817937737176681, −2.99967215008443706909046777920, −1.39768344702631611909790555926, 1.63638054258802604925139553035, 2.97124138444275376674572730431, 4.12387696835854151529776283449, 5.37378721469562050856578047924, 5.94490783502718610104148860057, 6.81146143695033815506916704088, 7.964597700592448465221653205898, 9.181481263265275782679351675548, 9.536626220199793931286843614478, 10.66662514296751376177831490135

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