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Properties

Label 2-33-11.9-c1-0-1
Degree $2$
Conductor $33$
Sign $0.836 - 0.548i$
Analytic cond. $0.263506$
Root an. cond. $0.513328$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.190 + 0.587i)2-s + (−0.809 + 0.587i)3-s + (1.30 + 0.951i)4-s + (−0.809 − 2.48i)5-s + (−0.190 − 0.587i)6-s + (−2.42 − 1.76i)7-s + (−1.80 + 1.31i)8-s + (0.309 − 0.951i)9-s + 1.61·10-s + (1.69 + 2.85i)11-s − 1.61·12-s + (0.545 − 1.67i)13-s + (1.5 − 1.08i)14-s + (2.11 + 1.53i)15-s + (0.572 + 1.76i)16-s + (0.5 + 1.53i)17-s + ⋯
L(s)  = 1  + (−0.135 + 0.415i)2-s + (−0.467 + 0.339i)3-s + (0.654 + 0.475i)4-s + (−0.361 − 1.11i)5-s + (−0.0779 − 0.239i)6-s + (−0.917 − 0.666i)7-s + (−0.639 + 0.464i)8-s + (0.103 − 0.317i)9-s + 0.511·10-s + (0.509 + 0.860i)11-s − 0.467·12-s + (0.151 − 0.465i)13-s + (0.400 − 0.291i)14-s + (0.546 + 0.397i)15-s + (0.143 + 0.440i)16-s + (0.121 + 0.373i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.836 - 0.548i$
Analytic conductor: \(0.263506\)
Root analytic conductor: \(0.513328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1/2),\ 0.836 - 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.605074 + 0.180815i\)
\(L(\frac12)\) \(\approx\) \(0.605074 + 0.180815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-1.69 - 2.85i)T \)
good2 \( 1 + (0.190 - 0.587i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (0.809 + 2.48i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.42 + 1.76i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.545 + 1.67i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.5 - 1.53i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.73 - 3.44i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 3.47T + 23T^{2} \)
29 \( 1 + (3.61 + 2.62i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.881 + 2.71i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.190 + 0.138i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-9.66 + 7.02i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 6.23T + 43T^{2} \)
47 \( 1 + (1.30 - 0.951i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.97 - 9.14i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.35 + 6.06i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.42 - 7.46i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 9.56T + 67T^{2} \)
71 \( 1 + (1.71 + 5.29i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.61 - 1.90i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.92 + 9.00i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.218 - 0.673i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 0.527T + 89T^{2} \)
97 \( 1 + (4.33 - 13.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−16.82983454225413192399787400211, −16.01576930751534649845596729771, −14.98636983243151817497475150590, −12.85915013788646699624350564553, −12.24047193025691409816799142577, −10.67312962199266996085002590042, −9.135631565494108772424258271424, −7.60256254186797441362040714584, −6.14354561328573155680917510391, −4.08107322614519672440380098907, 2.94245368695342572678930183445, 6.13043416985269818282051426535, 6.93042874817741200778147271705, 9.248471303729656249105420811613, 10.83054363188546449216534337670, 11.43972178294170517121552360496, 12.75305110198668963387882559563, 14.45286729621385068044881434016, 15.53265859262770130100391200284, 16.54632101849424605886363012987

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