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Properties

Label 2-33-11.10-c6-0-11
Degree $2$
Conductor $33$
Sign $-0.937 + 0.347i$
Analytic cond. $7.59178$
Root an. cond. $2.75531$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.49i·2-s + 15.5·3-s + 7.82·4-s − 189.·5-s − 116. i·6-s − 392. i·7-s − 538. i·8-s + 243·9-s + 1.42e3i·10-s + (−1.24e3 + 461. i)11-s + 121.·12-s − 3.16e3i·13-s − 2.94e3·14-s − 2.95e3·15-s − 3.53e3·16-s + 7.69e3i·17-s + ⋯
L(s)  = 1  − 0.936i·2-s + 0.577·3-s + 0.122·4-s − 1.51·5-s − 0.540i·6-s − 1.14i·7-s − 1.05i·8-s + 0.333·9-s + 1.42i·10-s + (−0.937 + 0.347i)11-s + 0.0705·12-s − 1.44i·13-s − 1.07·14-s − 0.876·15-s − 0.862·16-s + 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.937 + 0.347i$
Analytic conductor: \(7.59178\)
Root analytic conductor: \(2.75531\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :3),\ -0.937 + 0.347i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.244657 - 1.36596i\)
\(L(\frac12)\) \(\approx\) \(0.244657 - 1.36596i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 15.5T \)
11 \( 1 + (1.24e3 - 461. i)T \)
good2 \( 1 + 7.49iT - 64T^{2} \)
5 \( 1 + 189.T + 1.56e4T^{2} \)
7 \( 1 + 392. iT - 1.17e5T^{2} \)
13 \( 1 + 3.16e3iT - 4.82e6T^{2} \)
17 \( 1 - 7.69e3iT - 2.41e7T^{2} \)
19 \( 1 + 4.20e3iT - 4.70e7T^{2} \)
23 \( 1 - 1.53e4T + 1.48e8T^{2} \)
29 \( 1 + 1.78e4iT - 5.94e8T^{2} \)
31 \( 1 - 4.24e4T + 8.87e8T^{2} \)
37 \( 1 + 2.19e4T + 2.56e9T^{2} \)
41 \( 1 + 1.23e5iT - 4.75e9T^{2} \)
43 \( 1 - 9.56e4iT - 6.32e9T^{2} \)
47 \( 1 - 8.21e4T + 1.07e10T^{2} \)
53 \( 1 + 6.13e4T + 2.21e10T^{2} \)
59 \( 1 + 7.82e4T + 4.21e10T^{2} \)
61 \( 1 - 1.32e5iT - 5.15e10T^{2} \)
67 \( 1 - 9.08e4T + 9.04e10T^{2} \)
71 \( 1 - 1.70e5T + 1.28e11T^{2} \)
73 \( 1 - 1.69e5iT - 1.51e11T^{2} \)
79 \( 1 + 1.48e4iT - 2.43e11T^{2} \)
83 \( 1 + 4.38e5iT - 3.26e11T^{2} \)
89 \( 1 + 7.27e5T + 4.96e11T^{2} \)
97 \( 1 - 7.61e5T + 8.32e11T^{2} \)
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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

−15.23571492156157898583584631748, −13.24771226795581738287191317231, −12.43862878250300093403769279548, −10.93328737173467782922699547599, −10.31159877024315481677199320273, −8.146926287830974719900128308537, −7.23271170426593986992313855771, −4.18469403204910888853047962586, −3.02664307031210106553585887541, −0.68013833145883201412574428451, 2.80256667506157408310383699124, 4.95040704645644614494417329533, 6.89105263661248618759346563258, 7.988219569719050239307821286455, 8.964626340989319683166128830320, 11.29355447670366129747129745653, 12.12081726222936952028045208602, 13.99261751369953058308469708648, 15.17293286983627068953073893532, 15.76892983763185995695022727932

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