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Properties

Label 2-165-15.2-c1-0-12
Degree $2$
Conductor $165$
Sign $0.970 + 0.241i$
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  + (0.178 − 0.178i)2-s + (1.57 − 0.720i)3-s + 1.93i·4-s + (−0.754 − 2.10i)5-s + (0.152 − 0.410i)6-s + (1.41 + 1.41i)7-s + (0.704 + 0.704i)8-s + (1.96 − 2.26i)9-s + (−0.511 − 0.241i)10-s − i·11-s + (1.39 + 3.04i)12-s + (0.319 − 0.319i)13-s + 0.504·14-s + (−2.70 − 2.77i)15-s − 3.61·16-s + (−5.29 + 5.29i)17-s + ⋯
L(s)  = 1  + (0.126 − 0.126i)2-s + (0.909 − 0.415i)3-s + 0.967i·4-s + (−0.337 − 0.941i)5-s + (0.0624 − 0.167i)6-s + (0.533 + 0.533i)7-s + (0.248 + 0.248i)8-s + (0.654 − 0.756i)9-s + (−0.161 − 0.0764i)10-s − 0.301i·11-s + (0.402 + 0.880i)12-s + (0.0886 − 0.0886i)13-s + 0.134·14-s + (−0.698 − 0.715i)15-s − 0.904·16-s + (−1.28 + 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.970 + 0.241i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.970 + 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52434 - 0.187140i\)
\(L(\frac12)\) \(\approx\) \(1.52434 - 0.187140i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.57 + 0.720i)T \)
5 \( 1 + (0.754 + 2.10i)T \)
11 \( 1 + iT \)
good2 \( 1 + (-0.178 + 0.178i)T - 2iT^{2} \)
7 \( 1 + (-1.41 - 1.41i)T + 7iT^{2} \)
13 \( 1 + (-0.319 + 0.319i)T - 13iT^{2} \)
17 \( 1 + (5.29 - 5.29i)T - 17iT^{2} \)
19 \( 1 + 3.21iT - 19T^{2} \)
23 \( 1 + (0.692 + 0.692i)T + 23iT^{2} \)
29 \( 1 - 3.53T + 29T^{2} \)
31 \( 1 + 5.27T + 31T^{2} \)
37 \( 1 + (1.58 + 1.58i)T + 37iT^{2} \)
41 \( 1 - 7.02iT - 41T^{2} \)
43 \( 1 + (4.68 - 4.68i)T - 43iT^{2} \)
47 \( 1 + (2.12 - 2.12i)T - 47iT^{2} \)
53 \( 1 + (5.15 + 5.15i)T + 53iT^{2} \)
59 \( 1 - 7.26T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 + (-0.284 - 0.284i)T + 67iT^{2} \)
71 \( 1 + 12.3iT - 71T^{2} \)
73 \( 1 + (-5.26 + 5.26i)T - 73iT^{2} \)
79 \( 1 - 6.88iT - 79T^{2} \)
83 \( 1 + (-11.2 - 11.2i)T + 83iT^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 + (-8.85 - 8.85i)T + 97iT^{2} \)
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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

−12.97223281325021638434845838224, −12.04457025150381066412280765151, −11.12337927711715792738687590788, −9.275779663796847943659112372895, −8.439969511155608975273420143909, −8.069467085694517031451458430855, −6.65260259871744155034816932968, −4.76567927505792893753643376122, −3.64413530550974455381220226227, −2.07836876797892084006525475363, 2.19268023476175520704371875553, 3.89132713478073088663273922032, 5.01436456562460803995298172159, 6.72934598170933176987629277916, 7.53422423358228289589400937276, 8.887496142782630790413800111190, 9.993273321905889474355798949359, 10.67300429706299241293348434680, 11.56910524324823181644044733469, 13.34563997720670698967459699037

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