Properties

Label 2-33-33.32-c7-0-15
Degree $2$
Conductor $33$
Sign $0.789 + 0.613i$
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  − 4.00·2-s + (34.4 + 31.5i)3-s − 111.·4-s − 109. i·5-s + (−138. − 126. i)6-s − 1.22e3i·7-s + 960.·8-s + (192. + 2.17e3i)9-s + 437. i·10-s + (4.40e3 − 355. i)11-s + (−3.86e3 − 3.53e3i)12-s − 2.12e3i·13-s + 4.89e3i·14-s + (3.45e3 − 3.77e3i)15-s + 1.04e4·16-s + 2.38e4·17-s + ⋯
L(s)  = 1  − 0.353·2-s + (0.737 + 0.675i)3-s − 0.874·4-s − 0.391i·5-s + (−0.260 − 0.238i)6-s − 1.34i·7-s + 0.663·8-s + (0.0879 + 0.996i)9-s + 0.138i·10-s + (0.996 − 0.0804i)11-s + (−0.645 − 0.590i)12-s − 0.267i·13-s + 0.476i·14-s + (0.264 − 0.288i)15-s + 0.640·16-s + 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.789 + 0.613i$
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: \((2,\ 33,\ (\ :7/2),\ 0.789 + 0.613i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.43148 - 0.490977i\)
\(L(\frac12)\) \(\approx\) \(1.43148 - 0.490977i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-34.4 - 31.5i)T \)
11 \( 1 + (-4.40e3 + 355. i)T \)
good2 \( 1 + 4.00T + 128T^{2} \)
5 \( 1 + 109. iT - 7.81e4T^{2} \)
7 \( 1 + 1.22e3iT - 8.23e5T^{2} \)
13 \( 1 + 2.12e3iT - 6.27e7T^{2} \)
17 \( 1 - 2.38e4T + 4.10e8T^{2} \)
19 \( 1 + 4.67e4iT - 8.93e8T^{2} \)
23 \( 1 + 4.79e4iT - 3.40e9T^{2} \)
29 \( 1 + 1.69e5T + 1.72e10T^{2} \)
31 \( 1 + 7.43e4T + 2.75e10T^{2} \)
37 \( 1 - 2.77e5T + 9.49e10T^{2} \)
41 \( 1 + 1.45e5T + 1.94e11T^{2} \)
43 \( 1 - 2.64e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.20e6iT - 5.06e11T^{2} \)
53 \( 1 - 1.78e6iT - 1.17e12T^{2} \)
59 \( 1 - 3.88e5iT - 2.48e12T^{2} \)
61 \( 1 + 2.73e6iT - 3.14e12T^{2} \)
67 \( 1 + 4.55e5T + 6.06e12T^{2} \)
71 \( 1 - 2.64e6iT - 9.09e12T^{2} \)
73 \( 1 + 2.47e6iT - 1.10e13T^{2} \)
79 \( 1 - 4.90e6iT - 1.92e13T^{2} \)
83 \( 1 - 2.93e6T + 2.71e13T^{2} \)
89 \( 1 - 2.83e6iT - 4.42e13T^{2} \)
97 \( 1 - 6.00e6T + 8.07e13T^{2} \)
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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

−14.81615887326491790733128578379, −13.92626817953228175592857667293, −12.98589370189710092257564615794, −10.84626722194754903648936295676, −9.718891574401261048948635769273, −8.770964139363841672297696263369, −7.44965151171327053393668812570, −4.80699195505542376774063798365, −3.69375783852385943275009860212, −0.861486670923231932241685557539, 1.55966900467007246557130897899, 3.53643139236822007008908452668, 5.84890302313285472117601854057, 7.66752281194962653067823761837, 8.865487117925465768642557108674, 9.707049040802470001812485978151, 11.87536592688004216333653990773, 12.85044867871167964250859214233, 14.34209251767031239066866141758, 14.79030378433692053632131933246

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