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Properties

Label 2-630-63.25-c1-0-1
Degree $2$
Conductor $630$
Sign $-0.0419 - 0.999i$
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 + (−1.71 + 0.263i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−1.71 + 0.263i)6-s + (−2.25 − 1.38i)7-s + 8-s + (2.86 − 0.901i)9-s + (−0.5 + 0.866i)10-s + (2.64 + 4.58i)11-s + (−1.71 + 0.263i)12-s + (−1.04 − 1.81i)13-s + (−2.25 − 1.38i)14-s + (0.628 − 1.61i)15-s + 16-s + (−2.66 + 4.61i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.988 + 0.151i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (−0.698 + 0.107i)6-s + (−0.853 − 0.521i)7-s + 0.353·8-s + (0.953 − 0.300i)9-s + (−0.158 + 0.273i)10-s + (0.797 + 1.38i)11-s + (−0.494 + 0.0759i)12-s + (−0.289 − 0.502i)13-s + (−0.603 − 0.368i)14-s + (0.162 − 0.416i)15-s + 0.250·16-s + (−0.646 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0419 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0419 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0419 - 0.999i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.0419 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.861099 + 0.898046i\)
\(L(\frac12)\) \(\approx\) \(0.861099 + 0.898046i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 - 0.263i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.25 + 1.38i)T \)
good11 \( 1 + (-2.64 - 4.58i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.04 + 1.81i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.66 - 4.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.21 - 7.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.71 - 4.69i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.582 + 1.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.77T + 31T^{2} \)
37 \( 1 + (-1.59 - 2.76i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.51 + 2.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.77 - 3.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.36T + 47T^{2} \)
53 \( 1 + (-3.71 + 6.43i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 + 14.7T + 61T^{2} \)
67 \( 1 - 8.39T + 67T^{2} \)
71 \( 1 - 2.28T + 71T^{2} \)
73 \( 1 + (-6.16 + 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.25T + 79T^{2} \)
83 \( 1 + (3.46 - 5.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.37 - 11.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.09 + 5.35i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.83027837003594573714465569532, −10.09108497938674171523834921353, −9.569776672870095871947575894021, −7.72829673656790851958544857191, −7.04131342116159844941580140836, −6.24943040333545160181526220790, −5.43090652217777009085253477692, −4.11449122782156666451769650519, −3.64378448043775029152630596312, −1.69953233617346435560046415669, 0.61964427194700892466653667420, 2.59750540128330887085112449554, 3.89940173762379611937462321659, 4.95310095697873802539226546242, 5.76532659750587959966627680468, 6.63920638760071362207606137057, 7.25160789020532131161911533265, 8.852217466139569365320461165899, 9.420642801904222166043361167709, 10.73448521511950823981356920419

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