Properties

Label 2-33-1.1-c5-0-1
Degree $2$
Conductor $33$
Sign $1$
Analytic cond. $5.29266$
Root an. cond. $2.30057$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 9·3-s − 28·4-s + 46·5-s + 18·6-s + 148·7-s + 120·8-s + 81·9-s − 92·10-s + 121·11-s + 252·12-s + 574·13-s − 296·14-s − 414·15-s + 656·16-s − 722·17-s − 162·18-s + 2.16e3·19-s − 1.28e3·20-s − 1.33e3·21-s − 242·22-s − 2.53e3·23-s − 1.08e3·24-s − 1.00e3·25-s − 1.14e3·26-s − 729·27-s − 4.14e3·28-s + ⋯
L(s)  = 1  − 0.353·2-s − 0.577·3-s − 7/8·4-s + 0.822·5-s + 0.204·6-s + 1.14·7-s + 0.662·8-s + 1/3·9-s − 0.290·10-s + 0.301·11-s + 0.505·12-s + 0.942·13-s − 0.403·14-s − 0.475·15-s + 0.640·16-s − 0.605·17-s − 0.117·18-s + 1.37·19-s − 0.720·20-s − 0.659·21-s − 0.106·22-s − 0.999·23-s − 0.382·24-s − 0.322·25-s − 0.333·26-s − 0.192·27-s − 0.998·28-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: yes
Arithmetic: yes
Character: $\chi_{33} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.176604191\)
\(L(\frac12)\) \(\approx\) \(1.176604191\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + p^{2} T \)
11 \( 1 - p^{2} T \)
good2 \( 1 + p T + p^{5} T^{2} \)
5 \( 1 - 46 T + p^{5} T^{2} \)
7 \( 1 - 148 T + p^{5} T^{2} \)
13 \( 1 - 574 T + p^{5} T^{2} \)
17 \( 1 + 722 T + p^{5} T^{2} \)
19 \( 1 - 2160 T + p^{5} T^{2} \)
23 \( 1 + 2536 T + p^{5} T^{2} \)
29 \( 1 - 4650 T + p^{5} T^{2} \)
31 \( 1 - 5032 T + p^{5} T^{2} \)
37 \( 1 - 8118 T + p^{5} T^{2} \)
41 \( 1 + 5138 T + p^{5} T^{2} \)
43 \( 1 - 8304 T + p^{5} T^{2} \)
47 \( 1 - 24728 T + p^{5} T^{2} \)
53 \( 1 + 28746 T + p^{5} T^{2} \)
59 \( 1 + 5860 T + p^{5} T^{2} \)
61 \( 1 + 53658 T + p^{5} T^{2} \)
67 \( 1 - 30908 T + p^{5} T^{2} \)
71 \( 1 + 69648 T + p^{5} T^{2} \)
73 \( 1 + 18446 T + p^{5} T^{2} \)
79 \( 1 + 25300 T + p^{5} T^{2} \)
83 \( 1 + 17556 T + p^{5} T^{2} \)
89 \( 1 - 132570 T + p^{5} T^{2} \)
97 \( 1 - 70658 T + p^{5} T^{2} \)
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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

−15.90030767269105705842286664249, −14.15947957636882431597188319922, −13.52218881054187531776200162044, −11.85003592538449612068029697553, −10.53751455228906256242226959762, −9.313272384197628648374815930076, −7.971399737162129553526268175387, −5.91031575310589564754799983038, −4.52220299211329893679080348766, −1.27590322593231934581374699241, 1.27590322593231934581374699241, 4.52220299211329893679080348766, 5.91031575310589564754799983038, 7.971399737162129553526268175387, 9.313272384197628648374815930076, 10.53751455228906256242226959762, 11.85003592538449612068029697553, 13.52218881054187531776200162044, 14.15947957636882431597188319922, 15.90030767269105705842286664249

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