Properties

Label 2-33-1.1-c5-0-4
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  + 9.34·2-s − 9·3-s + 55.3·4-s + 69.4·5-s − 84.1·6-s + 8.69·7-s + 218.·8-s + 81·9-s + 649.·10-s + 121·11-s − 498.·12-s − 970.·13-s + 81.2·14-s − 625.·15-s + 268.·16-s − 424.·17-s + 757.·18-s − 1.43e3·19-s + 3.84e3·20-s − 78.2·21-s + 1.13e3·22-s + 2.85e3·23-s − 1.96e3·24-s + 1.69e3·25-s − 9.07e3·26-s − 729·27-s + 481.·28-s + ⋯
L(s)  = 1  + 1.65·2-s − 0.577·3-s + 1.72·4-s + 1.24·5-s − 0.953·6-s + 0.0670·7-s + 1.20·8-s + 0.333·9-s + 2.05·10-s + 0.301·11-s − 0.998·12-s − 1.59·13-s + 0.110·14-s − 0.717·15-s + 0.261·16-s − 0.356·17-s + 0.550·18-s − 0.909·19-s + 2.14·20-s − 0.0387·21-s + 0.498·22-s + 1.12·23-s − 0.695·24-s + 0.543·25-s − 2.63·26-s − 0.192·27-s + 0.115·28-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

\(L(3)\) \(\approx\) \(3.427396878\)
\(L(\frac12)\) \(\approx\) \(3.427396878\)
\(L(\frac{7}{2})\) 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 \( 1 - 9.34T + 32T^{2} \)
5 \( 1 - 69.4T + 3.12e3T^{2} \)
7 \( 1 - 8.69T + 1.68e4T^{2} \)
13 \( 1 + 970.T + 3.71e5T^{2} \)
17 \( 1 + 424.T + 1.41e6T^{2} \)
19 \( 1 + 1.43e3T + 2.47e6T^{2} \)
23 \( 1 - 2.85e3T + 6.43e6T^{2} \)
29 \( 1 + 7.46e3T + 2.05e7T^{2} \)
31 \( 1 - 1.03e4T + 2.86e7T^{2} \)
37 \( 1 - 167.T + 6.93e7T^{2} \)
41 \( 1 - 5.68e3T + 1.15e8T^{2} \)
43 \( 1 - 2.11e4T + 1.47e8T^{2} \)
47 \( 1 + 9.78e3T + 2.29e8T^{2} \)
53 \( 1 - 2.56e4T + 4.18e8T^{2} \)
59 \( 1 + 2.34e4T + 7.14e8T^{2} \)
61 \( 1 - 1.85e4T + 8.44e8T^{2} \)
67 \( 1 - 3.94e4T + 1.35e9T^{2} \)
71 \( 1 - 3.28e3T + 1.80e9T^{2} \)
73 \( 1 - 2.95e4T + 2.07e9T^{2} \)
79 \( 1 + 1.02e4T + 3.07e9T^{2} \)
83 \( 1 + 3.83e4T + 3.93e9T^{2} \)
89 \( 1 + 2.31e3T + 5.58e9T^{2} \)
97 \( 1 + 8.18e4T + 8.58e9T^{2} \)
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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

−15.23431867127850081414487985713, −14.34243030527849428448463305240, −13.22545181873379446567013817213, −12.39609326600468949754693355121, −11.08882777826823977393015283963, −9.614806171845037458059377460363, −6.88320511046792337825395795697, −5.73734338741906870614759001660, −4.60762522429610294231293191605, −2.35251158114376928065764876110, 2.35251158114376928065764876110, 4.60762522429610294231293191605, 5.73734338741906870614759001660, 6.88320511046792337825395795697, 9.614806171845037458059377460363, 11.08882777826823977393015283963, 12.39609326600468949754693355121, 13.22545181873379446567013817213, 14.34243030527849428448463305240, 15.23431867127850081414487985713

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