Properties

Label 2-33-1.1-c5-0-5
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  + 9.37·2-s + 9·3-s + 55.8·4-s + 0.277·5-s + 84.3·6-s − 105.·7-s + 223.·8-s + 81·9-s + 2.59·10-s − 121·11-s + 502.·12-s + 147.·13-s − 984.·14-s + 2.49·15-s + 307.·16-s − 1.43e3·17-s + 759.·18-s + 2.03e3·19-s + 15.4·20-s − 945.·21-s − 1.13e3·22-s + 828.·23-s + 2.01e3·24-s − 3.12e3·25-s + 1.38e3·26-s + 729·27-s − 5.86e3·28-s + ⋯
L(s)  = 1  + 1.65·2-s + 0.577·3-s + 1.74·4-s + 0.00495·5-s + 0.956·6-s − 0.810·7-s + 1.23·8-s + 0.333·9-s + 0.00821·10-s − 0.301·11-s + 1.00·12-s + 0.242·13-s − 1.34·14-s + 0.00286·15-s + 0.299·16-s − 1.20·17-s + 0.552·18-s + 1.29·19-s + 0.00865·20-s − 0.467·21-s − 0.499·22-s + 0.326·23-s + 0.712·24-s − 0.999·25-s + 0.401·26-s + 0.192·27-s − 1.41·28-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

\(L(3)\) \(\approx\) \(3.869507390\)
\(L(\frac12)\) \(\approx\) \(3.869507390\)
\(L(\frac{7}{2})\) 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 \( 1 - 9.37T + 32T^{2} \)
5 \( 1 - 0.277T + 3.12e3T^{2} \)
7 \( 1 + 105.T + 1.68e4T^{2} \)
13 \( 1 - 147.T + 3.71e5T^{2} \)
17 \( 1 + 1.43e3T + 1.41e6T^{2} \)
19 \( 1 - 2.03e3T + 2.47e6T^{2} \)
23 \( 1 - 828.T + 6.43e6T^{2} \)
29 \( 1 - 4.63e3T + 2.05e7T^{2} \)
31 \( 1 + 9.83e3T + 2.86e7T^{2} \)
37 \( 1 - 7.13e3T + 6.93e7T^{2} \)
41 \( 1 - 1.82e4T + 1.15e8T^{2} \)
43 \( 1 - 1.38e4T + 1.47e8T^{2} \)
47 \( 1 - 2.29e4T + 2.29e8T^{2} \)
53 \( 1 - 1.43e4T + 4.18e8T^{2} \)
59 \( 1 + 7.08e3T + 7.14e8T^{2} \)
61 \( 1 + 1.84e4T + 8.44e8T^{2} \)
67 \( 1 - 1.62e4T + 1.35e9T^{2} \)
71 \( 1 - 2.81e4T + 1.80e9T^{2} \)
73 \( 1 + 3.93e4T + 2.07e9T^{2} \)
79 \( 1 + 4.12e4T + 3.07e9T^{2} \)
83 \( 1 + 2.33e4T + 3.93e9T^{2} \)
89 \( 1 + 1.03e5T + 5.58e9T^{2} \)
97 \( 1 + 1.49e5T + 8.58e9T^{2} \)
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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

−15.50465083808900918992923520188, −14.21788260677073977346304498900, −13.36877286391376741900515804397, −12.51282127289383014778852372457, −11.08628983909474475216419951988, −9.307421501695817151644839556853, −7.25040400667280707185793066363, −5.80663917185888314720906691634, −4.09020043256355535371557988808, −2.70176463886111205392393810501, 2.70176463886111205392393810501, 4.09020043256355535371557988808, 5.80663917185888314720906691634, 7.25040400667280707185793066363, 9.307421501695817151644839556853, 11.08628983909474475216419951988, 12.51282127289383014778852372457, 13.36877286391376741900515804397, 14.21788260677073977346304498900, 15.50465083808900918992923520188

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