Properties

Label 2-33-11.2-c6-0-4
Degree $2$
Conductor $33$
Sign $-0.302 + 0.953i$
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  + (−9.54 − 3.10i)2-s + (−12.6 + 9.16i)3-s + (29.7 + 21.6i)4-s + (33.3 + 102. i)5-s + (148. − 48.3i)6-s + (−90.4 + 124. i)7-s + (160. + 220. i)8-s + (75.0 − 231. i)9-s − 1.08e3i·10-s + (−37.8 − 1.33e3i)11-s − 573.·12-s + (−3.18e3 − 1.03e3i)13-s + (1.25e3 − 908. i)14-s + (−1.35e3 − 987. i)15-s + (−1.57e3 − 4.84e3i)16-s + (4.52e3 − 1.47e3i)17-s + ⋯
L(s)  = 1  + (−1.19 − 0.387i)2-s + (−0.467 + 0.339i)3-s + (0.465 + 0.337i)4-s + (0.266 + 0.819i)5-s + (0.689 − 0.223i)6-s + (−0.263 + 0.363i)7-s + (0.313 + 0.431i)8-s + (0.103 − 0.317i)9-s − 1.08i·10-s + (−0.0284 − 0.999i)11-s − 0.331·12-s + (−1.45 − 0.471i)13-s + (0.455 − 0.331i)14-s + (−0.402 − 0.292i)15-s + (−0.384 − 1.18i)16-s + (0.921 − 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.302 + 0.953i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.302 + 0.953i$
Analytic conductor: \(7.59178\)
Root analytic conductor: \(2.75531\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :3),\ -0.302 + 0.953i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.219202 - 0.299679i\)
\(L(\frac12)\) \(\approx\) \(0.219202 - 0.299679i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (12.6 - 9.16i)T \)
11 \( 1 + (37.8 + 1.33e3i)T \)
good2 \( 1 + (9.54 + 3.10i)T + (51.7 + 37.6i)T^{2} \)
5 \( 1 + (-33.3 - 102. i)T + (-1.26e4 + 9.18e3i)T^{2} \)
7 \( 1 + (90.4 - 124. i)T + (-3.63e4 - 1.11e5i)T^{2} \)
13 \( 1 + (3.18e3 + 1.03e3i)T + (3.90e6 + 2.83e6i)T^{2} \)
17 \( 1 + (-4.52e3 + 1.47e3i)T + (1.95e7 - 1.41e7i)T^{2} \)
19 \( 1 + (3.07e3 + 4.22e3i)T + (-1.45e7 + 4.47e7i)T^{2} \)
23 \( 1 - 8.17e3T + 1.48e8T^{2} \)
29 \( 1 + (3.25e3 - 4.48e3i)T + (-1.83e8 - 5.65e8i)T^{2} \)
31 \( 1 + (-6.18e3 + 1.90e4i)T + (-7.18e8 - 5.21e8i)T^{2} \)
37 \( 1 + (1.28e4 + 9.33e3i)T + (7.92e8 + 2.44e9i)T^{2} \)
41 \( 1 + (7.85e4 + 1.08e5i)T + (-1.46e9 + 4.51e9i)T^{2} \)
43 \( 1 + 1.11e5iT - 6.32e9T^{2} \)
47 \( 1 + (-7.57e4 + 5.50e4i)T + (3.33e9 - 1.02e10i)T^{2} \)
53 \( 1 + (3.77e4 - 1.16e5i)T + (-1.79e10 - 1.30e10i)T^{2} \)
59 \( 1 + (3.32e4 + 2.41e4i)T + (1.30e10 + 4.01e10i)T^{2} \)
61 \( 1 + (-4.42e4 + 1.43e4i)T + (4.16e10 - 3.02e10i)T^{2} \)
67 \( 1 + 3.81e5T + 9.04e10T^{2} \)
71 \( 1 + (-9.27e4 - 2.85e5i)T + (-1.03e11 + 7.52e10i)T^{2} \)
73 \( 1 + (8.40e4 - 1.15e5i)T + (-4.67e10 - 1.43e11i)T^{2} \)
79 \( 1 + (8.09e4 + 2.63e4i)T + (1.96e11 + 1.42e11i)T^{2} \)
83 \( 1 + (-7.89e5 + 2.56e5i)T + (2.64e11 - 1.92e11i)T^{2} \)
89 \( 1 + 1.04e6T + 4.96e11T^{2} \)
97 \( 1 + (-2.41e5 + 7.43e5i)T + (-6.73e11 - 4.89e11i)T^{2} \)
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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

−15.24419438912622086185045437011, −13.99967967132325477719618802305, −12.12884630033184881504438285542, −10.84550835909525082817873253207, −10.08487182270640388607731876753, −8.891685397299496822224729840978, −7.22773291119416827900926615608, −5.41640957701878430972487875720, −2.72325398668177141270611294581, −0.32744676577824853317596657883, 1.38007224771226957119627233505, 4.79263151587166814580377771403, 6.79859015396010945385460371650, 7.909283532211829327453246971486, 9.421524089073025458517273302476, 10.24950751083272570063087842931, 12.19429816415786643333605478377, 13.07768655956826559136546291280, 14.84994932420700305132625423681, 16.55986994747516193457430126772

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