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Properties

Label 2-663-221.16-c1-0-34
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: $\chi_{663} (16, \cdot )$
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

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

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