↑

Properties

Label 2-630-35.17-c1-0-10
Degree $2$
Conductor $630$
Sign $0.699 + 0.714i$
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (2.21 − 0.318i)5-s + (0.559 + 2.58i)7-s + (−0.707 + 0.707i)8-s + (0.264 − 2.22i)10-s + (1.83 − 3.17i)11-s + (0.830 + 0.830i)13-s + (2.64 + 0.128i)14-s + (0.500 + 0.866i)16-s + (0.204 + 0.761i)17-s + (1.09 + 1.89i)19-s + (−2.07 − 0.830i)20-s + (−2.59 − 2.59i)22-s + (4.54 + 1.21i)23-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.433 − 0.249i)4-s + (0.989 − 0.142i)5-s + (0.211 + 0.977i)7-s + (−0.249 + 0.249i)8-s + (0.0837 − 0.702i)10-s + (0.553 − 0.958i)11-s + (0.230 + 0.230i)13-s + (0.706 + 0.0343i)14-s + (0.125 + 0.216i)16-s + (0.0494 + 0.184i)17-s + (0.251 + 0.434i)19-s + (−0.464 − 0.185i)20-s + (−0.553 − 0.553i)22-s + (0.947 + 0.253i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.699 + 0.714i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.699 + 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80110 - 0.757265i\)
\(L(\frac12)\) \(\approx\) \(1.80110 - 0.757265i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 \)
5 \( 1 + (-2.21 + 0.318i)T \)
7 \( 1 + (-0.559 - 2.58i)T \)
good11 \( 1 + (-1.83 + 3.17i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.830 - 0.830i)T + 13iT^{2} \)
17 \( 1 + (-0.204 - 0.761i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.09 - 1.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.54 - 1.21i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.62iT - 29T^{2} \)
31 \( 1 + (-0.0359 - 0.0207i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0664 + 0.248i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 8.98iT - 41T^{2} \)
43 \( 1 + (0.474 - 0.474i)T - 43iT^{2} \)
47 \( 1 + (6.18 + 1.65i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.04 + 7.64i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.35 - 9.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.72 + 0.996i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.39 + 1.71i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.11T + 71T^{2} \)
73 \( 1 + (-9.52 + 2.55i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (11.6 - 6.70i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.73 - 9.73i)T + 83iT^{2} \)
89 \( 1 + (0.715 + 1.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.16 - 3.16i)T - 97iT^{2} \)
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

−10.57321982463222272486725441919, −9.539860903111666606236435420610, −8.990051588494096893660901291593, −8.221719342393890830716389249385, −6.60776861976920372003873584553, −5.76499499807415918247369428303, −5.08409434094876073213135739066, −3.65081917622691334520659520703, −2.51927265441559047764650928932, −1.37076850971702774257066682277, 1.38823176069122681008577388684, 3.07240424579058203183003957990, 4.44218616682470715556874290073, 5.17444934628780316203575295140, 6.42884873099634419472710107699, 6.96884551161253358853336433277, 7.87828061712256657154867168899, 9.062256699275087851449663902830, 9.726391406396311777524436151067, 10.55256970225642962055337224443

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