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Properties

Label 2-663-13.4-c1-0-9
Degree $2$
Conductor $663$
Sign $-0.914 - 0.405i$
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  + (2.12 + 1.22i)2-s + (−0.5 + 0.866i)3-s + (2.01 + 3.49i)4-s − 0.284i·5-s + (−2.12 + 1.22i)6-s + (−4.07 + 2.35i)7-s + 5.00i·8-s + (−0.499 − 0.866i)9-s + (0.349 − 0.604i)10-s + (0.407 + 0.235i)11-s − 4.03·12-s + (2.08 + 2.94i)13-s − 11.5·14-s + (0.246 + 0.142i)15-s + (−2.10 + 3.65i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (1.50 + 0.868i)2-s + (−0.288 + 0.499i)3-s + (1.00 + 1.74i)4-s − 0.127i·5-s + (−0.868 + 0.501i)6-s + (−1.53 + 0.888i)7-s + 1.76i·8-s + (−0.166 − 0.288i)9-s + (0.110 − 0.191i)10-s + (0.122 + 0.0709i)11-s − 1.16·12-s + (0.578 + 0.815i)13-s − 3.08·14-s + (0.0635 + 0.0367i)15-s + (−0.527 + 0.913i)16-s + (−0.121 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(663\)    =    \(3 \cdot 13 \cdot 17\)
Sign: $-0.914 - 0.405i$
Analytic conductor: \(5.29408\)
Root analytic conductor: \(2.30088\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{663} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 663,\ (\ :1/2),\ -0.914 - 0.405i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.527668 + 2.49055i\)
\(L(\frac12)\) \(\approx\) \(0.527668 + 2.49055i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-2.08 - 2.94i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-2.12 - 1.22i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 0.284iT - 5T^{2} \)
7 \( 1 + (4.07 - 2.35i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.407 - 0.235i)T + (5.5 + 9.52i)T^{2} \)
19 \( 1 + (-0.117 + 0.0680i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.31 - 7.47i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.26 + 5.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.60iT - 31T^{2} \)
37 \( 1 + (-5.09 - 2.94i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.20 - 1.27i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.30 - 5.71i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.30iT - 47T^{2} \)
53 \( 1 + 7.22T + 53T^{2} \)
59 \( 1 + (-1.81 + 1.05i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.60 - 2.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.37 - 4.83i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.9 + 6.89i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.728iT - 73T^{2} \)
79 \( 1 + 15.1T + 79T^{2} \)
83 \( 1 + 13.0iT - 83T^{2} \)
89 \( 1 + (-3.80 - 2.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.5 - 6.10i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.36508768105622229116450358128, −9.810744007017887617491940852880, −9.283273317173127634951654846252, −8.068810120225426644063657978150, −6.82953994030914114892668535435, −6.18508469108127687721289033429, −5.64848939069337730462880580973, −4.49549195141445299275438400555, −3.66011766800054457460666257671, −2.71126391848698565626773651209, 0.883066928800285025186214397191, 2.61418884024243244603527187308, 3.44515386172925533751508575577, 4.33174334223386563943187913890, 5.56068092387949206904536328129, 6.43085322264813727466104281507, 6.89785383650877045135082029162, 8.373133274204966017334121227412, 9.774697949830794700348394526352, 10.71623034158369975656515351466

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