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Properties

Label 2-165-15.8-c1-0-9
Degree $2$
Conductor $165$
Sign $0.314 - 0.949i$
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  + (1.31 + 1.31i)2-s + (1.52 + 0.825i)3-s + 1.47i·4-s + (−1.49 − 1.66i)5-s + (0.918 + 3.09i)6-s + (−2.09 + 2.09i)7-s + (0.695 − 0.695i)8-s + (1.63 + 2.51i)9-s + (0.228 − 4.16i)10-s − i·11-s + (−1.21 + 2.24i)12-s + (0.161 + 0.161i)13-s − 5.51·14-s + (−0.897 − 3.76i)15-s + 4.77·16-s + (−3.37 − 3.37i)17-s + ⋯
L(s)  = 1  + (0.931 + 0.931i)2-s + (0.879 + 0.476i)3-s + 0.735i·4-s + (−0.667 − 0.744i)5-s + (0.375 + 1.26i)6-s + (−0.791 + 0.791i)7-s + (0.245 − 0.245i)8-s + (0.545 + 0.837i)9-s + (0.0721 − 1.31i)10-s − 0.301i·11-s + (−0.350 + 0.647i)12-s + (0.0448 + 0.0448i)13-s − 1.47·14-s + (−0.231 − 0.972i)15-s + 1.19·16-s + (−0.818 − 0.818i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.314 - 0.949i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.314 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60469 + 1.15914i\)
\(L(\frac12)\) \(\approx\) \(1.60469 + 1.15914i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.52 - 0.825i)T \)
5 \( 1 + (1.49 + 1.66i)T \)
11 \( 1 + iT \)
good2 \( 1 + (-1.31 - 1.31i)T + 2iT^{2} \)
7 \( 1 + (2.09 - 2.09i)T - 7iT^{2} \)
13 \( 1 + (-0.161 - 0.161i)T + 13iT^{2} \)
17 \( 1 + (3.37 + 3.37i)T + 17iT^{2} \)
19 \( 1 + 8.52iT - 19T^{2} \)
23 \( 1 + (3.15 - 3.15i)T - 23iT^{2} \)
29 \( 1 + 3.01T + 29T^{2} \)
31 \( 1 - 5.01T + 31T^{2} \)
37 \( 1 + (4.74 - 4.74i)T - 37iT^{2} \)
41 \( 1 - 4.49iT - 41T^{2} \)
43 \( 1 + (-0.336 - 0.336i)T + 43iT^{2} \)
47 \( 1 + (-6.15 - 6.15i)T + 47iT^{2} \)
53 \( 1 + (1.62 - 1.62i)T - 53iT^{2} \)
59 \( 1 + 5.16T + 59T^{2} \)
61 \( 1 + 3.39T + 61T^{2} \)
67 \( 1 + (-9.74 + 9.74i)T - 67iT^{2} \)
71 \( 1 + 2.23iT - 71T^{2} \)
73 \( 1 + (-5.43 - 5.43i)T + 73iT^{2} \)
79 \( 1 - 5.93iT - 79T^{2} \)
83 \( 1 + (0.365 - 0.365i)T - 83iT^{2} \)
89 \( 1 - 3.70T + 89T^{2} \)
97 \( 1 + (-4.94 + 4.94i)T - 97iT^{2} \)
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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

−13.37268651832860239604878079920, −12.48378788126502144176055814525, −11.21787802188179151026570469180, −9.579644690982394268610525121062, −8.888956750016994977918621365263, −7.74700631008119327928291997558, −6.63119018719156958129285070918, −5.19841871877601775646794394246, −4.34915768499635624026497800675, −3.01571106081857332206658906678, 2.16937882130937131302986705756, 3.61532460070629492148576233819, 4.00689206740429586353380623628, 6.28085266209515157041980337691, 7.41338166074995238490754695358, 8.349497971493801741539148197497, 10.07420012731094373553596468781, 10.64853450861240751759268999093, 12.05421289106303629159241090429, 12.57673104422850887135674256528

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