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Properties

Label 2-165-11.4-c1-0-0
Degree $2$
Conductor $165$
Sign $0.974 + 0.224i$
Analytic cond. $1.31753$
Root an. cond. $1.14783$
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  + (−1.50 − 1.09i)2-s + (−0.309 + 0.951i)3-s + (0.453 + 1.39i)4-s + (0.809 − 0.587i)5-s + (1.50 − 1.09i)6-s + (0.812 + 2.50i)7-s + (−0.306 + 0.943i)8-s + (−0.809 − 0.587i)9-s − 1.86·10-s + (1.77 − 2.80i)11-s − 1.46·12-s + (5.24 + 3.80i)13-s + (1.51 − 4.65i)14-s + (0.309 + 0.951i)15-s + (3.86 − 2.81i)16-s + (3.82 − 2.77i)17-s + ⋯
L(s)  = 1  + (−1.06 − 0.773i)2-s + (−0.178 + 0.549i)3-s + (0.226 + 0.697i)4-s + (0.361 − 0.262i)5-s + (0.614 − 0.446i)6-s + (0.307 + 0.945i)7-s + (−0.108 + 0.333i)8-s + (−0.269 − 0.195i)9-s − 0.588·10-s + (0.534 − 0.845i)11-s − 0.423·12-s + (1.45 + 1.05i)13-s + (0.404 − 1.24i)14-s + (0.0797 + 0.245i)15-s + (0.967 − 0.702i)16-s + (0.926 − 0.673i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.974 + 0.224i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.974 + 0.224i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.714489 - 0.0810869i\)
\(L(\frac12)\) \(\approx\) \(0.714489 - 0.0810869i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-1.77 + 2.80i)T \)
good2 \( 1 + (1.50 + 1.09i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (-0.812 - 2.50i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-5.24 - 3.80i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.82 + 2.77i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.164 + 0.506i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.54T + 23T^{2} \)
29 \( 1 + (-3.28 - 10.1i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.154 + 0.112i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.30 + 4.02i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.187 - 0.575i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9.24T + 43T^{2} \)
47 \( 1 + (2.72 - 8.39i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.41 + 3.93i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.83 + 5.63i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.60 + 1.89i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 5.22T + 67T^{2} \)
71 \( 1 + (-2.21 + 1.61i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.84 + 14.9i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.979 + 0.711i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.77 - 2.74i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 6.40T + 89T^{2} \)
97 \( 1 + (-5.92 - 4.30i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.25174198747081354697986496653, −11.51309130380318716265666427087, −10.82168390213741722554628722590, −9.633241496169282319644721799712, −8.931343302618508998772003816676, −8.321948271817456135558689309085, −6.26421854607025667634658842495, −5.19962789362856142310392042468, −3.33108392403209843912912580192, −1.54683384363687814863788143037, 1.25684603610817170149708097538, 3.81113269251069143033226947634, 5.88320921252065075802412162757, 6.71441675964304280063571250308, 7.81051628918365537717033464897, 8.368610544533841847082804430779, 9.928615990307537727218604782763, 10.41048737482491776571113334416, 11.80714302654651447207775824819, 12.99405204980600053257787419111

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