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Properties

Label 2-165-33.32-c5-0-16
Degree $2$
Conductor $165$
Sign $0.919 - 0.393i$
Analytic cond. $26.4633$
Root an. cond. $5.14425$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.04·2-s + (−14.9 − 4.57i)3-s + 17.5·4-s + 25i·5-s + (104. + 32.2i)6-s − 17.4i·7-s + 101.·8-s + (201. + 136. i)9-s − 176. i·10-s + (−259. − 306. i)11-s + (−262. − 80.5i)12-s − 30.7i·13-s + 122. i·14-s + (114. − 372. i)15-s − 1.27e3·16-s − 1.71e3·17-s + ⋯
L(s)  = 1  − 1.24·2-s + (−0.955 − 0.293i)3-s + 0.549·4-s + 0.447i·5-s + (1.19 + 0.365i)6-s − 0.134i·7-s + 0.560·8-s + (0.827 + 0.561i)9-s − 0.556i·10-s + (−0.646 − 0.763i)11-s + (−0.525 − 0.161i)12-s − 0.0505i·13-s + 0.167i·14-s + (0.131 − 0.427i)15-s − 1.24·16-s − 1.43·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.919 - 0.393i$
Analytic conductor: \(26.4633\)
Root analytic conductor: \(5.14425\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :5/2),\ 0.919 - 0.393i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.3679512932\)
\(L(\frac12)\) \(\approx\) \(0.3679512932\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (14.9 + 4.57i)T \)
5 \( 1 - 25iT \)
11 \( 1 + (259. + 306. i)T \)
good2 \( 1 + 7.04T + 32T^{2} \)
7 \( 1 + 17.4iT - 1.68e4T^{2} \)
13 \( 1 + 30.7iT - 3.71e5T^{2} \)
17 \( 1 + 1.71e3T + 1.41e6T^{2} \)
19 \( 1 - 1.78e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.96e3iT - 6.43e6T^{2} \)
29 \( 1 + 5.87e3T + 2.05e7T^{2} \)
31 \( 1 + 892.T + 2.86e7T^{2} \)
37 \( 1 - 5.77e3T + 6.93e7T^{2} \)
41 \( 1 - 1.46e3T + 1.15e8T^{2} \)
43 \( 1 - 5.21e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.06e4iT - 2.29e8T^{2} \)
53 \( 1 - 5.25e3iT - 4.18e8T^{2} \)
59 \( 1 - 2.80e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.86e4iT - 8.44e8T^{2} \)
67 \( 1 - 2.79e3T + 1.35e9T^{2} \)
71 \( 1 - 3.81e4iT - 1.80e9T^{2} \)
73 \( 1 + 4.22e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.42e4iT - 3.07e9T^{2} \)
83 \( 1 - 7.14e4T + 3.93e9T^{2} \)
89 \( 1 + 4.50e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.10e5T + 8.58e9T^{2} \)
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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

−11.56314393074805067888970273809, −10.71597086306235931813073785537, −10.26167070571244650965585038563, −8.893967325831210752688555621007, −7.87400731241775087853387024926, −6.91804779320836617779482496635, −5.81955264793483202997069931346, −4.32031141247103704405127899494, −2.13672906071006189006956740961, −0.58993268212944575365977879043, 0.39990773285708361159113189932, 1.88632998033623467936039444128, 4.33199274085924952553702152020, 5.30187514984662092519058779483, 6.83330248809197480123314401942, 7.75278877584339407910184870466, 9.184627523919937583979014975542, 9.597935305034397887233054833046, 10.85897234708067616338596841664, 11.37545442553881084093214740540

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