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Properties

Label 2-33-33.32-c7-0-24
Degree $2$
Conductor $33$
Sign $0.721 + 0.692i$
Analytic cond. $10.3087$
Root an. cond. $3.21071$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.1·2-s + (46.1 − 7.41i)3-s + 101.·4-s − 382. i·5-s + (698. − 112. i)6-s − 606. i·7-s − 408.·8-s + (2.07e3 − 684. i)9-s − 5.78e3i·10-s + (2.66e3 + 3.52e3i)11-s + (4.66e3 − 749. i)12-s + 1.19e4i·13-s − 9.17e3i·14-s + (−2.83e3 − 1.76e4i)15-s − 1.91e4·16-s + 1.15e4·17-s + ⋯
L(s)  = 1  + 1.33·2-s + (0.987 − 0.158i)3-s + 0.789·4-s − 1.36i·5-s + (1.32 − 0.212i)6-s − 0.667i·7-s − 0.281·8-s + (0.949 − 0.313i)9-s − 1.82i·10-s + (0.602 + 0.797i)11-s + (0.779 − 0.125i)12-s + 1.51i·13-s − 0.893i·14-s + (−0.216 − 1.35i)15-s − 1.16·16-s + 0.570·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.721 + 0.692i$
Analytic conductor: \(10.3087\)
Root analytic conductor: \(3.21071\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :7/2),\ 0.721 + 0.692i)\)

Particular Values

\(L(4)\) \(\approx\) \(4.08256 - 1.64124i\)
\(L(\frac12)\) \(\approx\) \(4.08256 - 1.64124i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-46.1 + 7.41i)T \)
11 \( 1 + (-2.66e3 - 3.52e3i)T \)
good2 \( 1 - 15.1T + 128T^{2} \)
5 \( 1 + 382. iT - 7.81e4T^{2} \)
7 \( 1 + 606. iT - 8.23e5T^{2} \)
13 \( 1 - 1.19e4iT - 6.27e7T^{2} \)
17 \( 1 - 1.15e4T + 4.10e8T^{2} \)
19 \( 1 - 3.96e4iT - 8.93e8T^{2} \)
23 \( 1 + 6.51e4iT - 3.40e9T^{2} \)
29 \( 1 - 1.06e5T + 1.72e10T^{2} \)
31 \( 1 + 5.03e4T + 2.75e10T^{2} \)
37 \( 1 + 4.87e5T + 9.49e10T^{2} \)
41 \( 1 + 3.86e5T + 1.94e11T^{2} \)
43 \( 1 - 3.85e5iT - 2.71e11T^{2} \)
47 \( 1 - 5.78e5iT - 5.06e11T^{2} \)
53 \( 1 + 4.24e5iT - 1.17e12T^{2} \)
59 \( 1 + 9.79e5iT - 2.48e12T^{2} \)
61 \( 1 + 3.28e6iT - 3.14e12T^{2} \)
67 \( 1 - 9.90e5T + 6.06e12T^{2} \)
71 \( 1 - 1.26e5iT - 9.09e12T^{2} \)
73 \( 1 - 1.59e6iT - 1.10e13T^{2} \)
79 \( 1 - 3.65e6iT - 1.92e13T^{2} \)
83 \( 1 + 9.71e6T + 2.71e13T^{2} \)
89 \( 1 + 1.86e6iT - 4.42e13T^{2} \)
97 \( 1 - 4.20e6T + 8.07e13T^{2} \)
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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

−14.43010967679884138163763539112, −13.98527716821944585835536988899, −12.67785888196736037732251762092, −12.13401624124270616955236792060, −9.682764301756310700217226623543, −8.516635031315033079540433860537, −6.75346703071811734504882640327, −4.70208783889704338792080185157, −3.86661470482666273427910810620, −1.62856223645928223958309169393, 2.78629932956906848668409684274, 3.47309888729137818980203846376, 5.51520259710312906380618594834, 7.04310619036881388601204631061, 8.791474443979983050394026999998, 10.45618694708569790709621016500, 11.88584375421553274230696879821, 13.33766523777937005176875222789, 14.10408415095967574816980141600, 15.14288170092452331252490421941

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