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Properties

Label 2-663-13.9-c1-0-15
Degree $2$
Conductor $663$
Sign $0.564 - 0.825i$
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  + (−0.903 + 1.56i)2-s + (−0.5 + 0.866i)3-s + (−0.630 − 1.09i)4-s + 0.935·5-s + (−0.903 − 1.56i)6-s + (−1.28 − 2.23i)7-s − 1.33·8-s + (−0.499 − 0.866i)9-s + (−0.844 + 1.46i)10-s + (3.23 − 5.60i)11-s + 1.26·12-s + (2.36 + 2.72i)13-s + 4.65·14-s + (−0.467 + 0.810i)15-s + (2.46 − 4.27i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.638 + 1.10i)2-s + (−0.288 + 0.499i)3-s + (−0.315 − 0.546i)4-s + 0.418·5-s + (−0.368 − 0.638i)6-s + (−0.487 − 0.843i)7-s − 0.471·8-s + (−0.166 − 0.288i)9-s + (−0.267 + 0.462i)10-s + (0.976 − 1.69i)11-s + 0.364·12-s + (0.655 + 0.754i)13-s + 1.24·14-s + (−0.120 + 0.209i)15-s + (0.616 − 1.06i)16-s + (−0.121 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(663\)    =    \(3 \cdot 13 \cdot 17\)
Sign: $0.564 - 0.825i$
Analytic conductor: \(5.29408\)
Root analytic conductor: \(2.30088\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{663} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 663,\ (\ :1/2),\ 0.564 - 0.825i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.881641 + 0.465090i\)
\(L(\frac12)\) \(\approx\) \(0.881641 + 0.465090i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.36 - 2.72i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.903 - 1.56i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 0.935T + 5T^{2} \)
7 \( 1 + (1.28 + 2.23i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.23 + 5.60i)T + (-5.5 - 9.52i)T^{2} \)
19 \( 1 + (-2.68 - 4.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.86 + 3.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.68 + 2.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.83T + 31T^{2} \)
37 \( 1 + (-0.193 + 0.334i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.73 + 4.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.51 + 2.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.45T + 47T^{2} \)
53 \( 1 + 4.08T + 53T^{2} \)
59 \( 1 + (-3.95 - 6.85i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.48 - 9.50i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.53 - 2.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.07 + 5.33i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.87T + 73T^{2} \)
79 \( 1 + 3.74T + 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 + (5.35 - 9.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.251 - 0.434i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.34592589413457468726039921497, −9.643894438457303051991341057781, −8.813723956518614287107365654541, −8.181958515896596404059149885223, −6.94603254732010889943593754952, −6.24838650180615682384247571125, −5.75311691518056179690463285825, −4.15451892937494246738176430394, −3.24237156613277830599261129624, −0.832162684501502802126109426939, 1.21187602286787938123862384941, 2.24239139611142699422001976538, 3.26479282970980597006536914527, 4.86307094111900039738864977418, 6.07526385294147499747587099810, 6.71239174734450498639253992530, 7.992958570811342294384687679240, 9.071549445505271300887401221256, 9.621768627238329997248136191556, 10.24774829636691513884152180530

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