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Properties

Label 2-663-221.152-c1-0-12
Degree $2$
Conductor $663$
Sign $0.274 - 0.961i$
Analytic cond. $5.29408$
Root an. cond. $2.30088$
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  + (0.207 − 0.358i)2-s + (0.866 + 0.5i)3-s + (0.914 + 1.58i)4-s − i·5-s + (0.358 − 0.207i)6-s + (−4.18 + 2.41i)7-s + 1.58·8-s + (0.499 + 0.866i)9-s + (−0.358 − 0.207i)10-s + (3.31 + 1.91i)11-s + 1.82i·12-s + (−1 − 3.46i)13-s + 2i·14-s + (0.5 − 0.866i)15-s + (−1.49 + 2.59i)16-s + (−1.18 + 3.94i)17-s + ⋯
L(s)  = 1  + (0.146 − 0.253i)2-s + (0.499 + 0.288i)3-s + (0.457 + 0.791i)4-s − 0.447i·5-s + (0.146 − 0.0845i)6-s + (−1.58 + 0.912i)7-s + 0.560·8-s + (0.166 + 0.288i)9-s + (−0.113 − 0.0654i)10-s + (0.999 + 0.577i)11-s + 0.527i·12-s + (−0.277 − 0.960i)13-s + 0.534i·14-s + (0.129 − 0.223i)15-s + (−0.374 + 0.649i)16-s + (−0.287 + 0.957i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(663\)    =    \(3 \cdot 13 \cdot 17\)
Sign: $0.274 - 0.961i$
Analytic conductor: \(5.29408\)
Root analytic conductor: \(2.30088\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{663} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 663,\ (\ :1/2),\ 0.274 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44608 + 1.09066i\)
\(L(\frac12)\) \(\approx\) \(1.44608 + 1.09066i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (1 + 3.46i)T \)
17 \( 1 + (1.18 - 3.94i)T \)
good2 \( 1 + (-0.207 + 0.358i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + iT - 5T^{2} \)
7 \( 1 + (4.18 - 2.41i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.31 - 1.91i)T + (5.5 + 9.52i)T^{2} \)
19 \( 1 + (-2.82 - 4.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.88 + 1.08i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.06 - 3.5i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.65iT - 31T^{2} \)
37 \( 1 + (7.64 + 4.41i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.19 + 3i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.91 + 5.04i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.31T + 47T^{2} \)
53 \( 1 - 9.65T + 53T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.36 + 4.82i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.08 + 3.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.79 + 5.65i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.17iT - 73T^{2} \)
79 \( 1 - 4.34iT - 79T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 + (-5.82 + 10.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.73 + i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.43420743012561567271110752245, −9.934020891058014158184828155579, −8.803235064303189706837938577080, −8.421651219628287646009285916340, −7.10416556429799642429398357833, −6.37782110400342472836799786082, −5.15569858667780967335820922617, −3.71066768286641323568515642021, −3.24031819602344041027589832196, −1.97853506071805036731909798340, 0.887489736774089390443411915707, 2.59975803254598113169296064067, 3.57502245792376631301848022344, 4.80275573625674714854981950363, 6.36024955956993115688634973565, 6.76236799021808170592531039999, 7.19394927258501358860675034792, 8.758113321793349543105447011077, 9.747878163418597730192297475224, 9.988614620996641008802126137695

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