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Properties

Label 2-33-33.32-c5-0-13
Degree $2$
Conductor $33$
Sign $0.106 + 0.994i$
Analytic cond. $5.29266$
Root an. cond. $2.30057$
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  + (15.5 − 1.65i)3-s − 32·4-s − 96.1i·5-s + (237.5 − 51.4i)9-s − 401. i·11-s + (−496 + 53.0i)12-s + (−159.5 − 1.49e3i)15-s + 1.02e3·16-s + 3.07e3i·20-s + 4.97e3i·23-s − 6.12e3·25-s + (3.59e3 − 1.19e3i)27-s + 7.77e3·31-s + (−665.5 − 6.22e3i)33-s + (−7.60e3 + 1.64e3i)36-s − 1.26e3·37-s + ⋯
L(s)  = 1  + (0.994 − 0.106i)3-s − 4-s − 1.72i·5-s + (0.977 − 0.211i)9-s − 1.00i·11-s + (−0.994 + 0.106i)12-s + (−0.183 − 1.71i)15-s + 16-s + 1.72i·20-s + 1.96i·23-s − 1.96·25-s + (0.949 − 0.314i)27-s + 1.45·31-s + (−0.106 − 0.994i)33-s + (−0.977 + 0.211i)36-s − 0.152·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.106 + 0.994i$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ 0.106 + 0.994i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.21693 - 1.09368i\)
\(L(\frac12)\) \(\approx\) \(1.21693 - 1.09368i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-15.5 + 1.65i)T \)
11 \( 1 + 401. iT \)
good2 \( 1 + 32T^{2} \)
5 \( 1 + 96.1iT - 3.12e3T^{2} \)
7 \( 1 - 1.68e4T^{2} \)
13 \( 1 - 3.71e5T^{2} \)
17 \( 1 + 1.41e6T^{2} \)
19 \( 1 - 2.47e6T^{2} \)
23 \( 1 - 4.97e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.05e7T^{2} \)
31 \( 1 - 7.77e3T + 2.86e7T^{2} \)
37 \( 1 + 1.26e3T + 6.93e7T^{2} \)
41 \( 1 + 1.15e8T^{2} \)
43 \( 1 - 1.47e8T^{2} \)
47 \( 1 + 1.75e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.14e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.73e4iT - 7.14e8T^{2} \)
61 \( 1 - 8.44e8T^{2} \)
67 \( 1 - 7.29e4T + 1.35e9T^{2} \)
71 \( 1 + 5.31e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.07e9T^{2} \)
79 \( 1 - 3.07e9T^{2} \)
83 \( 1 + 3.93e9T^{2} \)
89 \( 1 - 1.18e5iT - 5.58e9T^{2} \)
97 \( 1 + 1.63e5T + 8.58e9T^{2} \)
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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

−15.40478654736815424023554686792, −13.71817566051276078337238037839, −13.36704896334722217577147069374, −12.11955364246724241460641633706, −9.725057013617993165734644998908, −8.817082561242897380721959415442, −8.050547313759792387808329745339, −5.29633726447418472735087100040, −3.85653339879809286231674913238, −1.02269945630030399861323612426, 2.69402943737463911088975892599, 4.27470267460089352604975364464, 6.79509751801134785186566336813, 8.146454866320718588056049499674, 9.685361917310254111506945061205, 10.53635798038715966675450442312, 12.57658501569037454899175396393, 13.97790261706949847000263338902, 14.54788478364210083544831260402, 15.46586949043423593480956028835

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