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Properties

Label 2-630-15.8-c1-0-3
Degree $2$
Conductor $630$
Sign $0.0618 - 0.998i$
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 + 1.00i·4-s + (−2 − i)5-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (−0.707 − 2.12i)10-s + 3.41i·11-s + (4 + 4i)13-s + 1.00·14-s − 1.00·16-s + (3.41 + 3.41i)17-s + 2.82i·19-s + (1.00 − 2.00i)20-s + (−2.41 + 2.41i)22-s + (−0.828 + 0.828i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.894 − 0.447i)5-s + (0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s + (−0.223 − 0.670i)10-s + 1.02i·11-s + (1.10 + 1.10i)13-s + 0.267·14-s − 0.250·16-s + (0.828 + 0.828i)17-s + 0.648i·19-s + (0.223 − 0.447i)20-s + (−0.514 + 0.514i)22-s + (−0.172 + 0.172i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0618 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.0618 - 0.998i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.0618 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21114 + 1.13839i\)
\(L(\frac12)\) \(\approx\) \(1.21114 + 1.13839i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (2 + i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 - 3.41iT - 11T^{2} \)
13 \( 1 + (-4 - 4i)T + 13iT^{2} \)
17 \( 1 + (-3.41 - 3.41i)T + 17iT^{2} \)
19 \( 1 - 2.82iT - 19T^{2} \)
23 \( 1 + (0.828 - 0.828i)T - 23iT^{2} \)
29 \( 1 + 4.82T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + (2.24 - 2.24i)T - 37iT^{2} \)
41 \( 1 + 8.82iT - 41T^{2} \)
43 \( 1 + (8.07 + 8.07i)T + 43iT^{2} \)
47 \( 1 + (-0.757 - 0.757i)T + 47iT^{2} \)
53 \( 1 + (5.41 - 5.41i)T - 53iT^{2} \)
59 \( 1 + 2.82T + 59T^{2} \)
61 \( 1 - 9.89T + 61T^{2} \)
67 \( 1 + (-5.58 + 5.58i)T - 67iT^{2} \)
71 \( 1 - 6.82iT - 71T^{2} \)
73 \( 1 + (7.07 + 7.07i)T + 73iT^{2} \)
79 \( 1 - 5.65iT - 79T^{2} \)
83 \( 1 + (-4.82 + 4.82i)T - 83iT^{2} \)
89 \( 1 - 8.82T + 89T^{2} \)
97 \( 1 + (7.41 - 7.41i)T - 97iT^{2} \)
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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

−10.95174372088378946830238926025, −9.923667540233438628412441482875, −8.763785314818863153215659538553, −8.093247374626799464816880667178, −7.27228080476665279829224674074, −6.37170924010615488317665258565, −5.21380526244879577048842371195, −4.19592141619942338031065464718, −3.63859609125576287539812937809, −1.63878635833417879173299548236, 0.859180717129120512220510364139, 2.90143515600105378599693649810, 3.45686744822469589973206848687, 4.72497121321703070705734429907, 5.74030109427234935202162040552, 6.65505758318625086246569461189, 7.961522777508629907465368723883, 8.440063215837461594926550440546, 9.732166788234533835326455669917, 10.67526031546531814309637830001

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