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Properties

Label 2-630-315.178-c1-0-16
Degree $2$
Conductor $630$
Sign $0.423 - 0.905i$
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  + (−0.707 + 0.707i)2-s + (0.908 + 1.47i)3-s − 1.00i·4-s + (−0.622 − 2.14i)5-s + (−1.68 − 0.400i)6-s + (2.39 + 1.11i)7-s + (0.707 + 0.707i)8-s + (−1.35 + 2.67i)9-s + (1.95 + 1.07i)10-s + (−1.41 − 2.45i)11-s + (1.47 − 0.908i)12-s + (3.83 + 1.02i)13-s + (−2.48 + 0.903i)14-s + (2.60 − 2.86i)15-s − 1.00·16-s + (−1.00 + 0.270i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.524 + 0.851i)3-s − 0.500i·4-s + (−0.278 − 0.960i)5-s + (−0.687 − 0.163i)6-s + (0.906 + 0.423i)7-s + (0.250 + 0.250i)8-s + (−0.450 + 0.892i)9-s + (0.619 + 0.341i)10-s + (−0.426 − 0.738i)11-s + (0.425 − 0.262i)12-s + (1.06 + 0.284i)13-s + (−0.664 + 0.241i)14-s + (0.672 − 0.740i)15-s − 0.250·16-s + (−0.244 + 0.0655i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.423 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.423 - 0.905i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.423 - 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23272 + 0.784099i\)
\(L(\frac12)\) \(\approx\) \(1.23272 + 0.784099i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.908 - 1.47i)T \)
5 \( 1 + (0.622 + 2.14i)T \)
7 \( 1 + (-2.39 - 1.11i)T \)
good11 \( 1 + (1.41 + 2.45i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.83 - 1.02i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (1.00 - 0.270i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.63 - 2.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-8.78 + 2.35i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-6.84 - 3.95i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.07iT - 31T^{2} \)
37 \( 1 + (-4.85 - 1.30i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.73 + 3.30i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.98 - 1.60i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.30 - 1.30i)T + 47iT^{2} \)
53 \( 1 + (11.9 - 3.19i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 4.39iT - 61T^{2} \)
67 \( 1 + (-5.19 + 5.19i)T - 67iT^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 + (-0.670 - 2.50i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 - 1.96iT - 79T^{2} \)
83 \( 1 + (1.44 + 5.38i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-6.47 - 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.08 + 1.36i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.89340568481568784374976193756, −9.559754563015117348647578566552, −8.740441160575904760152762306274, −8.457477485460050316241337539851, −7.66166451299813060267775481408, −6.10329608299531330443424821462, −5.12439038921159024085975485265, −4.49848161998612558212048564650, −3.09270173031848956161658739994, −1.34743311134097991983405953578, 1.12285561147193410116747217405, 2.45863036282804162859851538967, 3.34733904081653682646033717198, 4.65789091628912773867210380842, 6.29544049575801423048500041657, 7.23439040171520341328967386111, 7.75384675512212340475952190125, 8.569834864842172112300853304941, 9.549342171331072571669954910106, 10.63119398006811503245859557771

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