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Properties

Label 2-33-33.32-c7-0-25
Degree $2$
Conductor $33$
Sign $-0.836 + 0.548i$
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  + 10.8·2-s + (−0.0827 − 46.7i)3-s − 9.86·4-s − 348. i·5-s + (−0.899 − 508. i)6-s + 1.04e3i·7-s − 1.49e3·8-s + (−2.18e3 + 7.74i)9-s − 3.78e3i·10-s + (−2.41e3 − 3.69e3i)11-s + (0.816 + 461. i)12-s − 2.64e3i·13-s + 1.13e4i·14-s + (−1.62e4 + 28.8i)15-s − 1.50e4·16-s + 3.03e4·17-s + ⋯
L(s)  = 1  + 0.960·2-s + (−0.00177 − 0.999i)3-s − 0.0770·4-s − 1.24i·5-s + (−0.00170 − 0.960i)6-s + 1.15i·7-s − 1.03·8-s + (−0.999 + 0.00354i)9-s − 1.19i·10-s + (−0.547 − 0.836i)11-s + (0.000136 + 0.0770i)12-s − 0.333i·13-s + 1.10i·14-s + (−1.24 + 0.00220i)15-s − 0.917·16-s + 1.49·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.836 + 0.548i$
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: \((2,\ 33,\ (\ :7/2),\ -0.836 + 0.548i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.506416 - 1.69454i\)
\(L(\frac12)\) \(\approx\) \(0.506416 - 1.69454i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0827 + 46.7i)T \)
11 \( 1 + (2.41e3 + 3.69e3i)T \)
good2 \( 1 - 10.8T + 128T^{2} \)
5 \( 1 + 348. iT - 7.81e4T^{2} \)
7 \( 1 - 1.04e3iT - 8.23e5T^{2} \)
13 \( 1 + 2.64e3iT - 6.27e7T^{2} \)
17 \( 1 - 3.03e4T + 4.10e8T^{2} \)
19 \( 1 + 3.68e4iT - 8.93e8T^{2} \)
23 \( 1 + 4.77e4iT - 3.40e9T^{2} \)
29 \( 1 + 1.89e5T + 1.72e10T^{2} \)
31 \( 1 - 2.69e5T + 2.75e10T^{2} \)
37 \( 1 + 9.05e4T + 9.49e10T^{2} \)
41 \( 1 - 4.78e5T + 1.94e11T^{2} \)
43 \( 1 - 1.67e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.50e5iT - 5.06e11T^{2} \)
53 \( 1 - 8.10e3iT - 1.17e12T^{2} \)
59 \( 1 + 1.25e6iT - 2.48e12T^{2} \)
61 \( 1 + 2.29e6iT - 3.14e12T^{2} \)
67 \( 1 + 3.88e6T + 6.06e12T^{2} \)
71 \( 1 - 6.05e5iT - 9.09e12T^{2} \)
73 \( 1 - 6.02e6iT - 1.10e13T^{2} \)
79 \( 1 + 1.99e6iT - 1.92e13T^{2} \)
83 \( 1 + 2.68e6T + 2.71e13T^{2} \)
89 \( 1 + 3.70e6iT - 4.42e13T^{2} \)
97 \( 1 + 4.64e6T + 8.07e13T^{2} \)
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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

−14.34991368761585002187061393140, −13.15870535544649897608140469952, −12.59227333621883690954968725317, −11.67136597464017098347281702726, −9.083871510763501822718544690883, −8.193478639065003975229111940803, −5.92030453712609348065426926831, −5.11543132042905392573324258484, −2.83527411790131286764820377420, −0.61069931325640018229414160225, 3.19249578093364114230811594337, 4.21218051180649434841660846771, 5.77628537581074711579298397063, 7.54144817079031741123341607194, 9.761734476795776940941296401480, 10.55047369001752123726121784976, 11.98607969437766385844699185943, 13.67152439460197797428347688233, 14.48054988260424067933788132179, 15.16837766745102965539338223786

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