Properties

Label 2-33-33.32-c5-0-9
Degree $2$
Conductor $33$
Sign $0.622 - 0.782i$
Analytic cond. $5.29266$
Root an. cond. $2.30057$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.1·2-s + (−10.9 + 11.0i)3-s + 70.2·4-s + 88.5i·5-s + (−110. + 112. i)6-s − 126. i·7-s + 387.·8-s + (−2.70 − 242. i)9-s + 895. i·10-s + (398. − 43.3i)11-s + (−770. + 779. i)12-s − 523. i·13-s − 1.27e3i·14-s + (−981. − 970. i)15-s + 1.66e3·16-s − 1.03e3·17-s + ⋯
L(s)  = 1  + 1.78·2-s + (−0.703 + 0.711i)3-s + 2.19·4-s + 1.58i·5-s + (−1.25 + 1.27i)6-s − 0.973i·7-s + 2.13·8-s + (−0.0111 − 0.999i)9-s + 2.83i·10-s + (0.994 − 0.108i)11-s + (−1.54 + 1.56i)12-s − 0.858i·13-s − 1.74i·14-s + (−1.12 − 1.11i)15-s + 1.62·16-s − 0.872·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.622 - 0.782i$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ 0.622 - 0.782i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.99626 + 1.44609i\)
\(L(\frac12)\) \(\approx\) \(2.99626 + 1.44609i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (10.9 - 11.0i)T \)
11 \( 1 + (-398. + 43.3i)T \)
good2 \( 1 - 10.1T + 32T^{2} \)
5 \( 1 - 88.5iT - 3.12e3T^{2} \)
7 \( 1 + 126. iT - 1.68e4T^{2} \)
13 \( 1 + 523. iT - 3.71e5T^{2} \)
17 \( 1 + 1.03e3T + 1.41e6T^{2} \)
19 \( 1 - 842. iT - 2.47e6T^{2} \)
23 \( 1 + 2.47e3iT - 6.43e6T^{2} \)
29 \( 1 - 3.56e3T + 2.05e7T^{2} \)
31 \( 1 + 215.T + 2.86e7T^{2} \)
37 \( 1 + 3.62e3T + 6.93e7T^{2} \)
41 \( 1 + 8.84e3T + 1.15e8T^{2} \)
43 \( 1 - 5.26e3iT - 1.47e8T^{2} \)
47 \( 1 - 9.05e3iT - 2.29e8T^{2} \)
53 \( 1 - 897. iT - 4.18e8T^{2} \)
59 \( 1 - 3.62e4iT - 7.14e8T^{2} \)
61 \( 1 + 1.53e4iT - 8.44e8T^{2} \)
67 \( 1 + 5.33e4T + 1.35e9T^{2} \)
71 \( 1 + 3.36e4iT - 1.80e9T^{2} \)
73 \( 1 + 7.61e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.06e5iT - 3.07e9T^{2} \)
83 \( 1 - 2.03e4T + 3.93e9T^{2} \)
89 \( 1 + 1.18e4iT - 5.58e9T^{2} \)
97 \( 1 + 6.77e4T + 8.58e9T^{2} \)
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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

−15.35368920387952205074067749018, −14.62984527508830163973880287648, −13.73746856960021850355706911644, −12.15392946308642109133032031882, −11.00352706699882174297256894160, −10.38816722121176336087405849143, −6.94623009680298540799411622938, −6.15708844386412759300434516557, −4.34062773245689249281738671893, −3.22105433788878974359730528812, 1.79621340541335969187927488482, 4.49681213509899616633546930376, 5.50700381132487095031050360260, 6.74316186577664261199465128014, 8.860608645640241041477123216766, 11.60010483670989608461209514354, 12.03353645717682260852145791606, 12.99773045784365004028839085533, 13.86246819926173729766047240680, 15.46954822180808959939348029388

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