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Properties

Label 2-33-11.10-c6-0-10
Degree $2$
Conductor $33$
Sign $-0.579 + 0.814i$
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  − 11.9i·2-s + 15.5·3-s − 78.6·4-s + 234.·5-s − 186. i·6-s − 368. i·7-s + 174. i·8-s + 243·9-s − 2.79e3i·10-s + (−771. + 1.08e3i)11-s − 1.22e3·12-s + 2.82e3i·13-s − 4.39e3·14-s + 3.65e3·15-s − 2.94e3·16-s − 4.47e3i·17-s + ⋯
L(s)  = 1  − 1.49i·2-s + 0.577·3-s − 1.22·4-s + 1.87·5-s − 0.861i·6-s − 1.07i·7-s + 0.340i·8-s + 0.333·9-s − 2.79i·10-s + (−0.579 + 0.814i)11-s − 0.709·12-s + 1.28i·13-s − 1.60·14-s + 1.08·15-s − 0.719·16-s − 0.910i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 + 0.814i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.579 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.579 + 0.814i$
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: \((2,\ 33,\ (\ :3),\ -0.579 + 0.814i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.10914 - 2.14966i\)
\(L(\frac12)\) \(\approx\) \(1.10914 - 2.14966i\)
\(L(4)\) 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 \( 1 + (771. - 1.08e3i)T \)
good2 \( 1 + 11.9iT - 64T^{2} \)
5 \( 1 - 234.T + 1.56e4T^{2} \)
7 \( 1 + 368. iT - 1.17e5T^{2} \)
13 \( 1 - 2.82e3iT - 4.82e6T^{2} \)
17 \( 1 + 4.47e3iT - 2.41e7T^{2} \)
19 \( 1 + 3.91e3iT - 4.70e7T^{2} \)
23 \( 1 + 9.08e3T + 1.48e8T^{2} \)
29 \( 1 - 2.50e4iT - 5.94e8T^{2} \)
31 \( 1 + 1.32e4T + 8.87e8T^{2} \)
37 \( 1 - 6.59e4T + 2.56e9T^{2} \)
41 \( 1 - 5.85e4iT - 4.75e9T^{2} \)
43 \( 1 - 6.51e4iT - 6.32e9T^{2} \)
47 \( 1 - 7.40e4T + 1.07e10T^{2} \)
53 \( 1 + 1.23e5T + 2.21e10T^{2} \)
59 \( 1 + 3.86e4T + 4.21e10T^{2} \)
61 \( 1 + 2.06e5iT - 5.15e10T^{2} \)
67 \( 1 - 1.63e5T + 9.04e10T^{2} \)
71 \( 1 + 2.58e5T + 1.28e11T^{2} \)
73 \( 1 - 2.42e5iT - 1.51e11T^{2} \)
79 \( 1 + 5.22e5iT - 2.43e11T^{2} \)
83 \( 1 - 1.02e6iT - 3.26e11T^{2} \)
89 \( 1 + 9.21e5T + 4.96e11T^{2} \)
97 \( 1 - 1.51e4T + 8.32e11T^{2} \)
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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

−14.25510014030674490126183438681, −13.60160808371222283379520752372, −12.74662129748922894252808392610, −10.99123694637843867660304584400, −9.863654414579287579010705424917, −9.360967715709214077651631133538, −6.90194080116224412475477699777, −4.57776221060929770589612168723, −2.58599372663401953198625448740, −1.47147528814225046857801491889, 2.38439482065377156074527019593, 5.61090938160960334961350781377, 5.99433713086058082358694589103, 8.029635811984696302227249026666, 9.012714240097765879840988590640, 10.30821223859741611799283452332, 12.87648056591547936047289869587, 13.78021077831366491411976340055, 14.75448007343311650141537527041, 15.69049099362269705919407593769

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