Properties

Label 2-33-1.1-c5-0-3
Degree $2$
Conductor $33$
Sign $-1$
Analytic cond. $5.29266$
Root an. cond. $2.30057$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.15·2-s − 9·3-s + 51.7·4-s + 95.5·5-s + 82.3·6-s − 209.·7-s − 180.·8-s + 81·9-s − 874.·10-s − 121·11-s − 465.·12-s − 335.·13-s + 1.91e3·14-s − 859.·15-s − 1.19·16-s − 799.·17-s − 741.·18-s − 658.·19-s + 4.94e3·20-s + 1.88e3·21-s + 1.10e3·22-s − 4.11e3·23-s + 1.62e3·24-s + 5.99e3·25-s + 3.07e3·26-s − 729·27-s − 1.08e4·28-s + ⋯
L(s)  = 1  − 1.61·2-s − 0.577·3-s + 1.61·4-s + 1.70·5-s + 0.934·6-s − 1.61·7-s − 0.999·8-s + 0.333·9-s − 2.76·10-s − 0.301·11-s − 0.933·12-s − 0.551·13-s + 2.61·14-s − 0.986·15-s − 0.00117·16-s − 0.670·17-s − 0.539·18-s − 0.418·19-s + 2.76·20-s + 0.933·21-s + 0.487·22-s − 1.62·23-s + 0.576·24-s + 1.91·25-s + 0.891·26-s − 0.192·27-s − 2.61·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-1$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9T \)
11 \( 1 + 121T \)
good2 \( 1 + 9.15T + 32T^{2} \)
5 \( 1 - 95.5T + 3.12e3T^{2} \)
7 \( 1 + 209.T + 1.68e4T^{2} \)
13 \( 1 + 335.T + 3.71e5T^{2} \)
17 \( 1 + 799.T + 1.41e6T^{2} \)
19 \( 1 + 658.T + 2.47e6T^{2} \)
23 \( 1 + 4.11e3T + 6.43e6T^{2} \)
29 \( 1 - 559.T + 2.05e7T^{2} \)
31 \( 1 + 6.05e3T + 2.86e7T^{2} \)
37 \( 1 + 1.40e4T + 6.93e7T^{2} \)
41 \( 1 - 1.84e3T + 1.15e8T^{2} \)
43 \( 1 - 1.62e3T + 1.47e8T^{2} \)
47 \( 1 - 2.07e4T + 2.29e8T^{2} \)
53 \( 1 + 7.58e3T + 4.18e8T^{2} \)
59 \( 1 - 1.84e4T + 7.14e8T^{2} \)
61 \( 1 + 1.69e4T + 8.44e8T^{2} \)
67 \( 1 + 5.61e3T + 1.35e9T^{2} \)
71 \( 1 - 3.70e3T + 1.80e9T^{2} \)
73 \( 1 + 1.98e4T + 2.07e9T^{2} \)
79 \( 1 - 6.40e4T + 3.07e9T^{2} \)
83 \( 1 + 4.63e4T + 3.93e9T^{2} \)
89 \( 1 + 5.39e4T + 5.58e9T^{2} \)
97 \( 1 - 1.45e5T + 8.58e9T^{2} \)
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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

−15.80915553911775117155067646633, −13.69607619656109181330460239955, −12.51797203314545027800527291420, −10.54104660329881261209451351712, −9.909161036763918665683027111396, −9.050934053527275779723401650026, −6.93799882683230437426918093224, −5.92849217179981441436906997408, −2.14021812507927974890337211927, 0, 2.14021812507927974890337211927, 5.92849217179981441436906997408, 6.93799882683230437426918093224, 9.050934053527275779723401650026, 9.909161036763918665683027111396, 10.54104660329881261209451351712, 12.51797203314545027800527291420, 13.69607619656109181330460239955, 15.80915553911775117155067646633

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