Properties

Label 2-33-1.1-c5-0-2
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  + 3.62·2-s + 9·3-s − 18.8·4-s + 57.7·5-s + 32.6·6-s + 251.·7-s − 184.·8-s + 81·9-s + 209.·10-s − 121·11-s − 169.·12-s − 277.·13-s + 910.·14-s + 519.·15-s − 66.1·16-s + 704.·17-s + 293.·18-s − 2.86e3·19-s − 1.08e3·20-s + 2.25e3·21-s − 438.·22-s − 1.06e3·23-s − 1.65e3·24-s + 206.·25-s − 1.00e3·26-s + 729·27-s − 4.73e3·28-s + ⋯
L(s)  = 1  + 0.641·2-s + 0.577·3-s − 0.588·4-s + 1.03·5-s + 0.370·6-s + 1.93·7-s − 1.01·8-s + 0.333·9-s + 0.662·10-s − 0.301·11-s − 0.339·12-s − 0.455·13-s + 1.24·14-s + 0.596·15-s − 0.0646·16-s + 0.591·17-s + 0.213·18-s − 1.81·19-s − 0.607·20-s + 1.11·21-s − 0.193·22-s − 0.420·23-s − 0.588·24-s + 0.0662·25-s − 0.292·26-s + 0.192·27-s − 1.14·28-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

\(L(3)\) \(\approx\) \(2.620263690\)
\(L(\frac12)\) \(\approx\) \(2.620263690\)
\(L(\frac{7}{2})\) 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 \( 1 - 3.62T + 32T^{2} \)
5 \( 1 - 57.7T + 3.12e3T^{2} \)
7 \( 1 - 251.T + 1.68e4T^{2} \)
13 \( 1 + 277.T + 3.71e5T^{2} \)
17 \( 1 - 704.T + 1.41e6T^{2} \)
19 \( 1 + 2.86e3T + 2.47e6T^{2} \)
23 \( 1 + 1.06e3T + 6.43e6T^{2} \)
29 \( 1 + 3.93e3T + 2.05e7T^{2} \)
31 \( 1 + 644.T + 2.86e7T^{2} \)
37 \( 1 + 9.04e3T + 6.93e7T^{2} \)
41 \( 1 - 1.82e4T + 1.15e8T^{2} \)
43 \( 1 + 4.05e3T + 1.47e8T^{2} \)
47 \( 1 - 2.07e4T + 2.29e8T^{2} \)
53 \( 1 + 2.64e4T + 4.18e8T^{2} \)
59 \( 1 - 4.29e3T + 7.14e8T^{2} \)
61 \( 1 + 6.83e3T + 8.44e8T^{2} \)
67 \( 1 + 5.67e4T + 1.35e9T^{2} \)
71 \( 1 - 3.18e3T + 1.80e9T^{2} \)
73 \( 1 + 7.39e3T + 2.07e9T^{2} \)
79 \( 1 - 2.43e4T + 3.07e9T^{2} \)
83 \( 1 - 1.02e5T + 3.93e9T^{2} \)
89 \( 1 - 4.95e4T + 5.58e9T^{2} \)
97 \( 1 + 9.22e4T + 8.58e9T^{2} \)
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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

−14.99325194156697193416206247668, −14.40524554217815183259477101427, −13.57594132238357617373735500931, −12.34007259110501297233343280900, −10.62439587140345802608846275568, −9.125902147853088621455115513635, −7.959609500874767055202244853026, −5.63390027193040633421602946195, −4.40693769843982712037763401810, −2.03685541957281458688932943194, 2.03685541957281458688932943194, 4.40693769843982712037763401810, 5.63390027193040633421602946195, 7.959609500874767055202244853026, 9.125902147853088621455115513635, 10.62439587140345802608846275568, 12.34007259110501297233343280900, 13.57594132238357617373735500931, 14.40524554217815183259477101427, 14.99325194156697193416206247668

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