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Properties

Label 2-33-33.32-c7-0-12
Degree $2$
Conductor $33$
Sign $0.999 + 0.0201i$
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  − 21.2·2-s + (13.2 + 44.8i)3-s + 324.·4-s − 116. i·5-s + (−282. − 954. i)6-s + 73.6i·7-s − 4.18e3·8-s + (−1.83e3 + 1.18e3i)9-s + 2.47e3i·10-s + (1.33e3 − 4.20e3i)11-s + (4.30e3 + 1.45e4i)12-s − 1.14e4i·13-s − 1.56e3i·14-s + (5.20e3 − 1.53e3i)15-s + 4.75e4·16-s + 1.92e4·17-s + ⋯
L(s)  = 1  − 1.88·2-s + (0.283 + 0.958i)3-s + 2.53·4-s − 0.415i·5-s + (−0.533 − 1.80i)6-s + 0.0812i·7-s − 2.89·8-s + (−0.839 + 0.544i)9-s + 0.781i·10-s + (0.302 − 0.953i)11-s + (0.719 + 2.43i)12-s − 1.44i·13-s − 0.152i·14-s + (0.398 − 0.117i)15-s + 2.90·16-s + 0.951·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0201i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.999 + 0.0201i$
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: \((2,\ 33,\ (\ :7/2),\ 0.999 + 0.0201i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.769585 - 0.00775775i\)
\(L(\frac12)\) \(\approx\) \(0.769585 - 0.00775775i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-13.2 - 44.8i)T \)
11 \( 1 + (-1.33e3 + 4.20e3i)T \)
good2 \( 1 + 21.2T + 128T^{2} \)
5 \( 1 + 116. iT - 7.81e4T^{2} \)
7 \( 1 - 73.6iT - 8.23e5T^{2} \)
13 \( 1 + 1.14e4iT - 6.27e7T^{2} \)
17 \( 1 - 1.92e4T + 4.10e8T^{2} \)
19 \( 1 - 4.25e4iT - 8.93e8T^{2} \)
23 \( 1 + 3.87e4iT - 3.40e9T^{2} \)
29 \( 1 - 1.15e5T + 1.72e10T^{2} \)
31 \( 1 - 1.57e5T + 2.75e10T^{2} \)
37 \( 1 + 2.05e5T + 9.49e10T^{2} \)
41 \( 1 - 4.57e5T + 1.94e11T^{2} \)
43 \( 1 + 2.15e5iT - 2.71e11T^{2} \)
47 \( 1 - 6.82e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.29e6iT - 1.17e12T^{2} \)
59 \( 1 - 2.61e6iT - 2.48e12T^{2} \)
61 \( 1 + 1.67e6iT - 3.14e12T^{2} \)
67 \( 1 - 2.45e6T + 6.06e12T^{2} \)
71 \( 1 + 6.58e5iT - 9.09e12T^{2} \)
73 \( 1 + 1.22e6iT - 1.10e13T^{2} \)
79 \( 1 + 6.39e6iT - 1.92e13T^{2} \)
83 \( 1 + 8.51e6T + 2.71e13T^{2} \)
89 \( 1 + 2.50e6iT - 4.42e13T^{2} \)
97 \( 1 - 5.77e6T + 8.07e13T^{2} \)
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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

−15.77240144002295191515069900028, −14.48907795285902198268307596223, −12.16152065910255749806718522049, −10.70058461883048907545619662139, −10.01667701278663102418490560948, −8.680353264388154783594494385307, −8.007745601950947078730110214356, −5.83207883349750093462384750946, −3.04485482271437128535378571711, −0.792199232016966766603938818618, 1.12352195097908296508357786168, 2.50245858042992821637399454775, 6.67069107823030520441261383932, 7.29746824587048571511308856977, 8.715567492933621801943338503109, 9.738465749162482773302048117243, 11.26501451240826616976670245381, 12.21496067476581684376400224569, 14.16855741536245264163650819236, 15.47445874383588896497085816220

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