↑

Properties

Label 2-630-15.2-c1-0-4
Degree $2$
Conductor $630$
Sign $0.920 - 0.391i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.920 - 0.391i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.920 - 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40658 + 0.286404i\)
\(L(\frac12)\) \(\approx\) \(1.40658 + 0.286404i\)
\(L(\frac{3}{2})\) 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 \( 1 + (7.41 + 7.41i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.33664592066108226229935687687, −9.836667081878072652997320747554, −8.603990876782233627498035649859, −8.421301579477715733648375806714, −7.01871557084179708342590129189, −6.19574283482600675327223362301, −5.33741629323801550905391353227, −4.37954327752272488647592107559, −2.53998568666270445304789557769, −1.24802337923093026931684889067, 1.25050269908082119073552964225, 2.53160967454764911143447382111, 3.67179433546089780236381731162, 4.94533003112060445076883295109, 6.32908503683765093839126747524, 6.78784412668583983455798591060, 8.282575210395348304920552734764, 8.789475643372092915084671226001, 9.809424803600791904890802246796, 10.48371280552734632101187963194

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