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Properties

Label 2-33-3.2-c8-0-0
Degree $2$
Conductor $33$
Sign $0.891 + 0.452i$
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  + 27.3i·2-s + (−36.6 + 72.2i)3-s − 489.·4-s − 131. i·5-s + (−1.97e3 − 1.00e3i)6-s − 44.0·7-s − 6.37e3i·8-s + (−3.87e3 − 5.29e3i)9-s + 3.58e3·10-s − 4.41e3i·11-s + (1.79e4 − 3.53e4i)12-s − 6.56e3·13-s − 1.20e3i·14-s + (9.48e3 + 4.81e3i)15-s + 4.87e4·16-s + 1.05e5i·17-s + ⋯
L(s)  = 1  + 1.70i·2-s + (−0.452 + 0.891i)3-s − 1.91·4-s − 0.210i·5-s + (−1.52 − 0.772i)6-s − 0.0183·7-s − 1.55i·8-s + (−0.589 − 0.807i)9-s + 0.358·10-s − 0.301i·11-s + (0.865 − 1.70i)12-s − 0.229·13-s − 0.0312i·14-s + (0.187 + 0.0951i)15-s + 0.743·16-s + 1.26i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.891 + 0.452i$
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: \((2,\ 33,\ (\ :4),\ 0.891 + 0.452i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0865241 - 0.0207134i\)
\(L(\frac12)\) \(\approx\) \(0.0865241 - 0.0207134i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (36.6 - 72.2i)T \)
11 \( 1 + 4.41e3iT \)
good2 \( 1 - 27.3iT - 256T^{2} \)
5 \( 1 + 131. iT - 3.90e5T^{2} \)
7 \( 1 + 44.0T + 5.76e6T^{2} \)
13 \( 1 + 6.56e3T + 8.15e8T^{2} \)
17 \( 1 - 1.05e5iT - 6.97e9T^{2} \)
19 \( 1 + 8.26e4T + 1.69e10T^{2} \)
23 \( 1 + 2.06e4iT - 7.83e10T^{2} \)
29 \( 1 + 1.04e6iT - 5.00e11T^{2} \)
31 \( 1 + 1.11e6T + 8.52e11T^{2} \)
37 \( 1 + 1.77e6T + 3.51e12T^{2} \)
41 \( 1 + 4.15e6iT - 7.98e12T^{2} \)
43 \( 1 - 6.06e5T + 1.16e13T^{2} \)
47 \( 1 - 7.81e6iT - 2.38e13T^{2} \)
53 \( 1 - 4.90e5iT - 6.22e13T^{2} \)
59 \( 1 + 1.98e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.67e7T + 1.91e14T^{2} \)
67 \( 1 - 1.41e7T + 4.06e14T^{2} \)
71 \( 1 - 1.70e7iT - 6.45e14T^{2} \)
73 \( 1 + 1.82e7T + 8.06e14T^{2} \)
79 \( 1 + 2.20e7T + 1.51e15T^{2} \)
83 \( 1 - 4.39e7iT - 2.25e15T^{2} \)
89 \( 1 - 9.75e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.33e8T + 7.83e15T^{2} \)
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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

−15.99726991943378916416194230909, −15.12896096539011613102303824561, −14.23951881073528006649242126295, −12.66818513271192160501555272225, −10.81142336608611640558530501716, −9.302632761127713564485758772336, −8.211861609275450818428772121091, −6.50642499515988127613077651136, −5.42219511749060235482567951022, −4.10800026994561736752131909970, 0.04022298422643556383225185484, 1.55788466648283283589960714416, 2.93666933076068036376827755945, 4.96544192498556250068882616177, 7.05572503725913839460942629463, 8.915881939669842014302842394114, 10.42221687182763166089987067469, 11.40393946320215294851290983005, 12.37942558091668309437022011950, 13.23251879729057760540728508618

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