Properties

Label 2-33-1.1-c5-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.34·2-s − 9·3-s + 37.6·4-s − 107.·5-s + 75.1·6-s − 26.6·7-s − 47.1·8-s + 81·9-s + 896.·10-s + 121·11-s − 338.·12-s + 904.·13-s + 222.·14-s + 967.·15-s − 811.·16-s − 495.·17-s − 676.·18-s − 1.50e3·19-s − 4.04e3·20-s + 240.·21-s − 1.00e3·22-s + 2.39e3·23-s + 424.·24-s + 8.42e3·25-s − 7.55e3·26-s − 729·27-s − 1.00e3·28-s + ⋯
L(s)  = 1  − 1.47·2-s − 0.577·3-s + 1.17·4-s − 1.92·5-s + 0.851·6-s − 0.205·7-s − 0.260·8-s + 0.333·9-s + 2.83·10-s + 0.301·11-s − 0.679·12-s + 1.48·13-s + 0.303·14-s + 1.10·15-s − 0.792·16-s − 0.415·17-s − 0.491·18-s − 0.954·19-s − 2.26·20-s + 0.118·21-s − 0.444·22-s + 0.943·23-s + 0.150·24-s + 2.69·25-s − 2.19·26-s − 0.192·27-s − 0.242·28-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: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.3484711604\)
\(L(\frac12)\) \(\approx\) \(0.3484711604\)
\(L(\frac{7}{2})\) 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 \( 1 + 8.34T + 32T^{2} \)
5 \( 1 + 107.T + 3.12e3T^{2} \)
7 \( 1 + 26.6T + 1.68e4T^{2} \)
13 \( 1 - 904.T + 3.71e5T^{2} \)
17 \( 1 + 495.T + 1.41e6T^{2} \)
19 \( 1 + 1.50e3T + 2.47e6T^{2} \)
23 \( 1 - 2.39e3T + 6.43e6T^{2} \)
29 \( 1 + 5.13e3T + 2.05e7T^{2} \)
31 \( 1 + 410.T + 2.86e7T^{2} \)
37 \( 1 - 5.82e3T + 6.93e7T^{2} \)
41 \( 1 - 1.85e4T + 1.15e8T^{2} \)
43 \( 1 + 788.T + 1.47e8T^{2} \)
47 \( 1 - 3.97e3T + 2.29e8T^{2} \)
53 \( 1 - 1.51e4T + 4.18e8T^{2} \)
59 \( 1 - 4.16e4T + 7.14e8T^{2} \)
61 \( 1 + 2.99e4T + 8.44e8T^{2} \)
67 \( 1 - 2.58e4T + 1.35e9T^{2} \)
71 \( 1 - 5.81e4T + 1.80e9T^{2} \)
73 \( 1 - 2.38e4T + 2.07e9T^{2} \)
79 \( 1 - 2.74e4T + 3.07e9T^{2} \)
83 \( 1 - 2.40e4T + 3.93e9T^{2} \)
89 \( 1 + 5.58e4T + 5.58e9T^{2} \)
97 \( 1 + 1.01e5T + 8.58e9T^{2} \)
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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

−16.07809785890652776194182721632, −15.17492946747649701884565699229, −12.84285927985097752328433191834, −11.28812293790970773653276316684, −10.94095646036259825092598392044, −9.008339541023446007286035294706, −7.987752016260584808050827269045, −6.76955182103526619177115805629, −4.06079175080597074273269044081, −0.70022603535538289271875179639, 0.70022603535538289271875179639, 4.06079175080597074273269044081, 6.76955182103526619177115805629, 7.987752016260584808050827269045, 9.008339541023446007286035294706, 10.94095646036259825092598392044, 11.28812293790970773653276316684, 12.84285927985097752328433191834, 15.17492946747649701884565699229, 16.07809785890652776194182721632

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