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Properties

Label 2-33-33.8-c1-0-1
Degree $2$
Conductor $33$
Sign $0.925 + 0.378i$
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  + (0.951 − 0.690i)2-s + (−1.72 − 0.0877i)3-s + (−0.190 + 0.587i)4-s + (0.224 − 0.309i)5-s + (−1.70 + 1.11i)6-s + (−2.92 − 0.951i)7-s + (0.951 + 2.92i)8-s + (2.98 + 0.303i)9-s − 0.449i·10-s + (1.31 − 3.04i)11-s + (0.381 − i)12-s + (0.427 + 0.587i)13-s + (−3.44 + 1.11i)14-s + (−0.415 + 0.514i)15-s + (1.92 + 1.40i)16-s + (−3.44 − 2.5i)17-s + ⋯
L(s)  = 1  + (0.672 − 0.488i)2-s + (−0.998 − 0.0506i)3-s + (−0.0954 + 0.293i)4-s + (0.100 − 0.138i)5-s + (−0.696 + 0.453i)6-s + (−1.10 − 0.359i)7-s + (0.336 + 1.03i)8-s + (0.994 + 0.101i)9-s − 0.141i·10-s + (0.396 − 0.918i)11-s + (0.110 − 0.288i)12-s + (0.118 + 0.163i)13-s + (−0.919 + 0.298i)14-s + (−0.107 + 0.132i)15-s + (0.481 + 0.350i)16-s + (−0.834 − 0.606i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.925 + 0.378i$
Analytic conductor: \(0.263506\)
Root analytic conductor: \(0.513328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1/2),\ 0.925 + 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.718462 - 0.141395i\)
\(L(\frac12)\) \(\approx\) \(0.718462 - 0.141395i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.0877i)T \)
11 \( 1 + (-1.31 + 3.04i)T \)
good2 \( 1 + (-0.951 + 0.690i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-0.224 + 0.309i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (2.92 + 0.951i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.427 - 0.587i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.44 + 2.5i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.5 + 0.812i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 6.23iT - 23T^{2} \)
29 \( 1 + (-0.726 + 2.23i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.73 - 3.44i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.927 + 2.85i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.224 + 0.690i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.88iT - 43T^{2} \)
47 \( 1 + (-7.91 + 2.57i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.08 + 1.5i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.363 - 0.118i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.54 + 2.12i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 7.32T + 67T^{2} \)
71 \( 1 + (3.13 - 4.30i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (8.94 + 2.90i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.33 - 3.21i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.11 - 3.71i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 0.527iT - 89T^{2} \)
97 \( 1 + (-3.54 + 2.57i)T + (29.9 - 92.2i)T^{2} \)
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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

−16.71598701585100330154426890754, −15.83272568578034673439612318555, −13.73172458165368866469515204442, −13.07322998099115866602821907639, −11.85481905701146750910478057394, −10.93885988255215580123500415287, −9.257735107787117260995632060917, −7.08301800112742237995953734903, −5.46600676539405139219776822096, −3.68156172421129633680571128073, 4.41845372521572954201232401543, 6.01459408397326209964519010137, 6.83402265617460341215744444414, 9.528982181714282366495144249913, 10.56502042330295727788460097488, 12.34454968185236291416625223570, 13.08983316747983387514878240873, 14.67022975066258374038319495365, 15.70401740790844847234767683839, 16.54149232261129993820536012945

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