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Properties

Label 2-165-5.4-c1-0-1
Degree $2$
Conductor $165$
Sign $-0.749 - 0.662i$
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  + 1.48i·2-s + i·3-s − 0.193·4-s + (−1.48 + 1.67i)5-s − 1.48·6-s − 1.19i·7-s + 2.67i·8-s − 9-s + (−2.48 − 2.19i)10-s + 11-s − 0.193i·12-s + 0.806i·13-s + 1.76·14-s + (−1.67 − 1.48i)15-s − 4.35·16-s − 3.76i·17-s + ⋯
L(s)  = 1  + 1.04i·2-s + 0.577i·3-s − 0.0969·4-s + (−0.662 + 0.749i)5-s − 0.604·6-s − 0.451i·7-s + 0.945i·8-s − 0.333·9-s + (−0.784 − 0.693i)10-s + 0.301·11-s − 0.0559i·12-s + 0.223i·13-s + 0.472·14-s + (−0.432 − 0.382i)15-s − 1.08·16-s − 0.913i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.749 - 0.662i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ -0.749 - 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.395829 + 1.04521i\)
\(L(\frac12)\) \(\approx\) \(0.395829 + 1.04521i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + (1.48 - 1.67i)T \)
11 \( 1 - T \)
good2 \( 1 - 1.48iT - 2T^{2} \)
7 \( 1 + 1.19iT - 7T^{2} \)
13 \( 1 - 0.806iT - 13T^{2} \)
17 \( 1 + 3.76iT - 17T^{2} \)
19 \( 1 - 5.35T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 4.31T + 29T^{2} \)
31 \( 1 - 0.962T + 31T^{2} \)
37 \( 1 + 1.61iT - 37T^{2} \)
41 \( 1 - 9.08T + 41T^{2} \)
43 \( 1 - 4.41iT - 43T^{2} \)
47 \( 1 + 12.3iT - 47T^{2} \)
53 \( 1 - 1.42iT - 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 + 0.0752T + 61T^{2} \)
67 \( 1 - 2.70iT - 67T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 + 10.7iT - 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 + 9.89iT - 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 + 11.4iT - 97T^{2} \)
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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

−13.78953006119763841521639059891, −11.86366906208083451104419412544, −11.31121787763888967858268662191, −10.19906073915398753106600901108, −8.987299273180007689724312625251, −7.66272035950727495763118504218, −7.09756438691985000544857407153, −5.86428820859869945912781221266, −4.52857750228403223662545657098, −3.05639885360294619392388421297, 1.21956741581243862017984637287, 2.90113401681275246849643611099, 4.31851509368664661542632254062, 5.95425153960378698893669772119, 7.32168950992682340257386495713, 8.442288094094856898763813947054, 9.476260888961579847257088249322, 10.73655288829009720883736592240, 11.68108486014377555521997134743, 12.35959905347996465801243796378

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