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Properties

Label 2-33-3.2-c4-0-10
Degree $2$
Conductor $33$
Sign $-0.615 + 0.788i$
Analytic cond. $3.41120$
Root an. cond. $1.84694$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.24i·2-s + (7.09 + 5.53i)3-s − 22.9·4-s − 42.0i·5-s + (34.5 − 44.2i)6-s − 11.7·7-s + 43.4i·8-s + (19.6 + 78.5i)9-s − 262.·10-s + 36.4i·11-s + (−162. − 127. i)12-s + 186.·13-s + 73.4i·14-s + (232. − 298. i)15-s − 96.1·16-s + 216. i·17-s + ⋯
L(s)  = 1  − 1.56i·2-s + (0.788 + 0.615i)3-s − 1.43·4-s − 1.68i·5-s + (0.960 − 1.23i)6-s − 0.240·7-s + 0.679i·8-s + (0.243 + 0.970i)9-s − 2.62·10-s + 0.301i·11-s + (−1.13 − 0.882i)12-s + 1.10·13-s + 0.374i·14-s + (1.03 − 1.32i)15-s − 0.375·16-s + 0.750i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.615 + 0.788i$
Analytic conductor: \(3.41120\)
Root analytic conductor: \(1.84694\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :2),\ -0.615 + 0.788i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.730800 - 1.49729i\)
\(L(\frac12)\) \(\approx\) \(0.730800 - 1.49729i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-7.09 - 5.53i)T \)
11 \( 1 - 36.4iT \)
good2 \( 1 + 6.24iT - 16T^{2} \)
5 \( 1 + 42.0iT - 625T^{2} \)
7 \( 1 + 11.7T + 2.40e3T^{2} \)
13 \( 1 - 186.T + 2.85e4T^{2} \)
17 \( 1 - 216. iT - 8.35e4T^{2} \)
19 \( 1 - 579.T + 1.30e5T^{2} \)
23 \( 1 + 266. iT - 2.79e5T^{2} \)
29 \( 1 - 775. iT - 7.07e5T^{2} \)
31 \( 1 - 902.T + 9.23e5T^{2} \)
37 \( 1 + 1.10e3T + 1.87e6T^{2} \)
41 \( 1 - 682. iT - 2.82e6T^{2} \)
43 \( 1 + 1.43e3T + 3.41e6T^{2} \)
47 \( 1 + 3.62e3iT - 4.87e6T^{2} \)
53 \( 1 - 987. iT - 7.89e6T^{2} \)
59 \( 1 - 1.77e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.84e3T + 1.38e7T^{2} \)
67 \( 1 + 4.89e3T + 2.01e7T^{2} \)
71 \( 1 + 519. iT - 2.54e7T^{2} \)
73 \( 1 - 7.82e3T + 2.83e7T^{2} \)
79 \( 1 - 416.T + 3.89e7T^{2} \)
83 \( 1 + 2.98e3iT - 4.74e7T^{2} \)
89 \( 1 + 3.34e3iT - 6.27e7T^{2} \)
97 \( 1 + 7.54e3T + 8.85e7T^{2} \)
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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

−15.66983047967880717253127991271, −13.73520500705011414118157071386, −12.97961160814724740500279422649, −11.88181659724564276024609127752, −10.35225478758894570185712129804, −9.280368160164325320333168122488, −8.424481628531028576749481351783, −4.88486486135562505300935594334, −3.54749921223883744287510051736, −1.40843251256726647481537590499, 3.20043040559172205757490383512, 6.11176873719731055491313276999, 7.04549917324986155593562851649, 8.002724380166032857478590542419, 9.565706711830710086307414353557, 11.44710193276788505423839319814, 13.75406807088554301677163303421, 13.96515086425928874578024518970, 15.24664859739589218752457696588, 15.91868805688073068997631739645

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