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Properties

Label 2-165-55.4-c1-0-3
Degree $2$
Conductor $165$
Sign $0.376 - 0.926i$
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  + (−1.12 + 1.54i)2-s + (−0.951 − 0.309i)3-s + (−0.506 − 1.55i)4-s + (2.14 − 0.621i)5-s + (1.54 − 1.12i)6-s + (4.23 − 1.37i)7-s + (−0.656 − 0.213i)8-s + (0.809 + 0.587i)9-s + (−1.44 + 4.01i)10-s + (−1.22 + 3.08i)11-s + 1.63i·12-s + (−0.313 + 0.432i)13-s + (−2.62 + 8.08i)14-s + (−2.23 − 0.0725i)15-s + (3.71 − 2.69i)16-s + (2.67 + 3.67i)17-s + ⋯
L(s)  = 1  + (−0.792 + 1.09i)2-s + (−0.549 − 0.178i)3-s + (−0.253 − 0.779i)4-s + (0.960 − 0.277i)5-s + (0.629 − 0.457i)6-s + (1.60 − 0.520i)7-s + (−0.231 − 0.0753i)8-s + (0.269 + 0.195i)9-s + (−0.458 + 1.26i)10-s + (−0.368 + 0.929i)11-s + 0.472i·12-s + (−0.0870 + 0.119i)13-s + (−0.702 + 2.16i)14-s + (−0.577 − 0.0187i)15-s + (0.928 − 0.674i)16-s + (0.647 + 0.891i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.376 - 0.926i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.376 - 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.698380 + 0.470123i\)
\(L(\frac12)\) \(\approx\) \(0.698380 + 0.470123i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (-2.14 + 0.621i)T \)
11 \( 1 + (1.22 - 3.08i)T \)
good2 \( 1 + (1.12 - 1.54i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 + (-4.23 + 1.37i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.313 - 0.432i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.67 - 3.67i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.594 - 1.83i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 5.18iT - 23T^{2} \)
29 \( 1 + (-1.46 - 4.52i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.82 + 2.05i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.501 - 0.162i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.45 + 4.46i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.02iT - 43T^{2} \)
47 \( 1 + (11.4 + 3.70i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.95 + 4.06i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.21 + 12.9i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (9.45 - 6.87i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 11.5iT - 67T^{2} \)
71 \( 1 + (2.72 - 1.97i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.618 + 0.200i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (14.3 + 10.4i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.49 + 3.43i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 8.24T + 89T^{2} \)
97 \( 1 + (0.669 - 0.920i)T + (-29.9 - 92.2i)T^{2} \)
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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

−12.99514418094565852497550287107, −12.08479585640639240163051558356, −10.67455914265042473690119359568, −9.961117961078987853584685247277, −8.628242714298835760608984846923, −7.82117443978411954556476458275, −6.83654399229565412740630442876, −5.67299097953208184679135993674, −4.70341955340694236386673583569, −1.63293575446630085190138968553, 1.45602688719259894630838786538, 2.86091778608742097153346549519, 5.08951801812347457873328519588, 5.92298173775289168665920815295, 7.78498718129015923425037873850, 8.911710177367422948105073198949, 9.787634301559457928052766740869, 10.83073995466235296161192348961, 11.32377312488223270583188520974, 12.12740037757304451168810996086

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