Properties

Label 2-165-165.32-c0-0-1
Degree $2$
Conductor $165$
Sign $0.525 + 0.850i$
Analytic cond. $0.0823457$
Root an. cond. $0.286959$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s i·4-s i·5-s + 9-s + i·11-s + i·12-s + i·15-s − 16-s − 20-s + (1 + i)23-s − 25-s − 27-s i·33-s i·36-s + (1 + i)37-s + ⋯
L(s)  = 1  − 3-s i·4-s i·5-s + 9-s + i·11-s + i·12-s + i·15-s − 16-s − 20-s + (1 + i)23-s − 25-s − 27-s i·33-s i·36-s + (1 + i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(0.0823457\)
Root analytic conductor: \(0.286959\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :0),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5124325930\)
\(L(\frac12)\) \(\approx\) \(0.5124325930\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + iT \)
11 \( 1 - iT \)
good2 \( 1 + iT^{2} \)
7 \( 1 - iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1 + i)T - iT^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 + 2T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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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

−12.80673196513357283655651651307, −11.89286158380736589932985726108, −10.95760162931962160367338185741, −9.886113795338133613131479961396, −9.212166649138198661442524128971, −7.52181963981657335249948391031, −6.29718563946434510881114313508, −5.22744274393448764590226605608, −4.50569616635841001221954384387, −1.47786184727082608092675371664, 2.92269017277789166739562281737, 4.28989820608368591912189901280, 5.92208702387101527522442263091, 6.88508674445321842543244130273, 7.83768670628207218172432513277, 9.206166716883822325673110415413, 10.68830450870167606505493878184, 11.16066340036910961031208830099, 12.14520824230585135033675814288, 13.04626153620637147925560241206

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