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Properties

Label 2-33-3.2-c8-0-2
Degree $2$
Conductor $33$
Sign $-0.865 - 0.500i$
Analytic cond. $13.4434$
Root an. cond. $3.66653$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.17i·2-s + (−40.5 + 70.1i)3-s + 189.·4-s + 920. i·5-s + (572. + 331. i)6-s − 1.09e3·7-s − 3.63e3i·8-s + (−3.26e3 − 5.68e3i)9-s + 7.51e3·10-s + 4.41e3i·11-s + (−7.67e3 + 1.32e4i)12-s − 2.57e4·13-s + 8.93e3i·14-s + (−6.45e4 − 3.73e4i)15-s + 1.87e4·16-s + 3.74e4i·17-s + ⋯
L(s)  = 1  − 0.510i·2-s + (−0.500 + 0.865i)3-s + 0.739·4-s + 1.47i·5-s + (0.441 + 0.255i)6-s − 0.455·7-s − 0.888i·8-s + (−0.498 − 0.867i)9-s + 0.751·10-s + 0.301i·11-s + (−0.370 + 0.639i)12-s − 0.902·13-s + 0.232i·14-s + (−1.27 − 0.737i)15-s + 0.285·16-s + 0.447i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.865 - 0.500i$
Analytic conductor: \(13.4434\)
Root analytic conductor: \(3.66653\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :4),\ -0.865 - 0.500i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.233203 + 0.868595i\)
\(L(\frac12)\) \(\approx\) \(0.233203 + 0.868595i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (40.5 - 70.1i)T \)
11 \( 1 - 4.41e3iT \)
good2 \( 1 + 8.17iT - 256T^{2} \)
5 \( 1 - 920. iT - 3.90e5T^{2} \)
7 \( 1 + 1.09e3T + 5.76e6T^{2} \)
13 \( 1 + 2.57e4T + 8.15e8T^{2} \)
17 \( 1 - 3.74e4iT - 6.97e9T^{2} \)
19 \( 1 + 1.91e5T + 1.69e10T^{2} \)
23 \( 1 - 2.57e5iT - 7.83e10T^{2} \)
29 \( 1 + 2.72e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.44e6T + 8.52e11T^{2} \)
37 \( 1 - 2.40e5T + 3.51e12T^{2} \)
41 \( 1 - 3.01e6iT - 7.98e12T^{2} \)
43 \( 1 - 6.24e6T + 1.16e13T^{2} \)
47 \( 1 + 3.04e6iT - 2.38e13T^{2} \)
53 \( 1 - 6.80e5iT - 6.22e13T^{2} \)
59 \( 1 - 1.81e7iT - 1.46e14T^{2} \)
61 \( 1 - 1.16e7T + 1.91e14T^{2} \)
67 \( 1 + 1.58e7T + 4.06e14T^{2} \)
71 \( 1 - 4.27e7iT - 6.45e14T^{2} \)
73 \( 1 - 1.96e7T + 8.06e14T^{2} \)
79 \( 1 - 4.04e7T + 1.51e15T^{2} \)
83 \( 1 - 4.13e7iT - 2.25e15T^{2} \)
89 \( 1 - 3.21e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.37e8T + 7.83e15T^{2} \)
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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

−15.23209932208531841150524811318, −14.71701042016575710412289974802, −12.61115214017099124515464778997, −11.33503249157477949996689548249, −10.57324624844955030034446988086, −9.696903254933945894369661225377, −7.16743117473204548655220404605, −6.08266916843451860897245812224, −3.79120142297967987720400787063, −2.48456516006523556043329019482, 0.36267380328321007906475418488, 2.09318868283253642273734894900, 5.03757398866462639651413775776, 6.28291828572131789637808728803, 7.59772067037359018833737578576, 8.856244160389461124444777154946, 10.88404537470365571397540493417, 12.30482519381278594575372213116, 12.82666221692377846218780089022, 14.43778301766748980706854842843

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