Properties

Label 2-33-3.2-c8-0-11
Degree $2$
Conductor $33$
Sign $-0.0218 - 0.999i$
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  + 9.43i·2-s + (80.9 − 1.77i)3-s + 167.·4-s + 654. i·5-s + (16.6 + 763. i)6-s − 1.00e3·7-s + 3.99e3i·8-s + (6.55e3 − 286. i)9-s − 6.17e3·10-s − 4.41e3i·11-s + (1.35e4 − 295. i)12-s + 1.56e4·13-s − 9.47e3i·14-s + (1.15e3 + 5.29e4i)15-s + 5.10e3·16-s + 2.86e4i·17-s + ⋯
L(s)  = 1  + 0.589i·2-s + (0.999 − 0.0218i)3-s + 0.652·4-s + 1.04i·5-s + (0.0128 + 0.589i)6-s − 0.418·7-s + 0.974i·8-s + (0.999 − 0.0436i)9-s − 0.617·10-s − 0.301i·11-s + (0.652 − 0.0142i)12-s + 0.549·13-s − 0.246i·14-s + (0.0228 + 1.04i)15-s + 0.0779·16-s + 0.343i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0218 - 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0218 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.0218 - 0.999i$
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: \((2,\ 33,\ (\ :4),\ -0.0218 - 0.999i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.03484 + 2.07981i\)
\(L(\frac12)\) \(\approx\) \(2.03484 + 2.07981i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-80.9 + 1.77i)T \)
11 \( 1 + 4.41e3iT \)
good2 \( 1 - 9.43iT - 256T^{2} \)
5 \( 1 - 654. iT - 3.90e5T^{2} \)
7 \( 1 + 1.00e3T + 5.76e6T^{2} \)
13 \( 1 - 1.56e4T + 8.15e8T^{2} \)
17 \( 1 - 2.86e4iT - 6.97e9T^{2} \)
19 \( 1 + 1.55e5T + 1.69e10T^{2} \)
23 \( 1 - 3.98e5iT - 7.83e10T^{2} \)
29 \( 1 + 8.41e5iT - 5.00e11T^{2} \)
31 \( 1 - 1.36e5T + 8.52e11T^{2} \)
37 \( 1 - 1.28e6T + 3.51e12T^{2} \)
41 \( 1 + 3.48e6iT - 7.98e12T^{2} \)
43 \( 1 - 1.46e6T + 1.16e13T^{2} \)
47 \( 1 + 4.25e6iT - 2.38e13T^{2} \)
53 \( 1 + 1.30e7iT - 6.22e13T^{2} \)
59 \( 1 - 2.01e6iT - 1.46e14T^{2} \)
61 \( 1 - 1.44e7T + 1.91e14T^{2} \)
67 \( 1 - 2.96e7T + 4.06e14T^{2} \)
71 \( 1 + 1.15e6iT - 6.45e14T^{2} \)
73 \( 1 + 4.15e7T + 8.06e14T^{2} \)
79 \( 1 + 1.18e7T + 1.51e15T^{2} \)
83 \( 1 + 8.83e7iT - 2.25e15T^{2} \)
89 \( 1 + 6.59e7iT - 3.93e15T^{2} \)
97 \( 1 + 9.98e7T + 7.83e15T^{2} \)
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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

−15.18784766845840859709728942184, −14.35976113920690821277669868441, −13.15207872941440816497763048531, −11.34370535874242238277469814317, −10.13899350811322942450396590077, −8.455143376718772382136313150210, −7.23722945002991523377739526378, −6.18516528351006095699605947749, −3.52073590197196507915072455834, −2.18230832893404733471237735502, 1.19558602180601278704790700304, 2.69294700521151377473858606468, 4.30731271723911924273991018513, 6.65856172767297937537304897071, 8.303471723296784407920876117534, 9.462528301692006582747218212755, 10.75324956656815289363727123213, 12.50317987903073102917139530277, 12.98930714901158305373594998239, 14.62518566176833343366854155467

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