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Properties

Label 2-630-105.89-c1-0-13
Degree $2$
Conductor $630$
Sign $-0.0237 + 0.999i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.27 − 1.83i)5-s + (−0.732 − 2.54i)7-s + 0.999·8-s + (0.948 + 2.02i)10-s + (2.07 − 1.19i)11-s − 5.67·13-s + (2.56 + 0.636i)14-s + (−0.5 + 0.866i)16-s + (−1.79 + 1.03i)17-s + (−5.12 − 2.95i)19-s + (−2.22 − 0.191i)20-s + 2.39i·22-s + (−0.930 + 1.61i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.572 − 0.820i)5-s + (−0.276 − 0.960i)7-s + 0.353·8-s + (0.299 + 0.640i)10-s + (0.625 − 0.361i)11-s − 1.57·13-s + (0.686 + 0.170i)14-s + (−0.125 + 0.216i)16-s + (−0.435 + 0.251i)17-s + (−1.17 − 0.678i)19-s + (−0.498 − 0.0427i)20-s + 0.511i·22-s + (−0.194 + 0.336i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0237 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0237 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0237 + 0.999i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.0237 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.614069 - 0.628839i\)
\(L(\frac12)\) \(\approx\) \(0.614069 - 0.628839i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-1.27 + 1.83i)T \)
7 \( 1 + (0.732 + 2.54i)T \)
good11 \( 1 + (-2.07 + 1.19i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.67T + 13T^{2} \)
17 \( 1 + (1.79 - 1.03i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.12 + 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.930 - 1.61i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.88iT - 29T^{2} \)
31 \( 1 + (3.92 - 2.26i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.57 - 1.48i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.04T + 41T^{2} \)
43 \( 1 + 8.55iT - 43T^{2} \)
47 \( 1 + (-4.83 - 2.78i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.09 + 3.62i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.00 - 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.7 - 6.22i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.60 + 3.81i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.14iT - 71T^{2} \)
73 \( 1 + (-0.541 - 0.937i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.38 + 14.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.6iT - 83T^{2} \)
89 \( 1 + (6.63 - 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−10.12250867376788089904760151108, −9.428029651070131854786566561989, −8.725033347312733512289889399141, −7.68260188707811664877674818167, −6.84026130701837336849900212928, −5.98427294504810316141731555677, −4.85166558594335080036060932075, −4.06721708406571691454560040372, −2.15043331780285416967961878794, −0.51128419289477635313660471699, 2.06618131500285113924309404095, 2.68439898532940445622697270584, 4.08499748984299443235162396566, 5.35776831935453408817325203841, 6.43830700935989245985273418088, 7.23812069343204961999016916796, 8.396128387067316186631467565920, 9.472724604423322857240948777602, 9.746695466482526990004103879446, 10.79878914274019039112666608798

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