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Properties

Label 2-33-33.32-c3-0-4
Degree $2$
Conductor $33$
Sign $0.934 - 0.355i$
Analytic cond. $1.94706$
Root an. cond. $1.39537$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.61·2-s + (4.07 + 3.22i)3-s − 1.15·4-s + 5.53i·5-s + (10.6 + 8.43i)6-s − 31.3i·7-s − 23.9·8-s + (6.22 + 26.2i)9-s + 14.4i·10-s + (18.7 − 31.3i)11-s + (−4.69 − 3.71i)12-s + 31.3i·13-s − 82.0i·14-s + (−17.8 + 22.5i)15-s − 53.4·16-s − 85.3·17-s + ⋯
L(s)  = 1  + 0.925·2-s + (0.784 + 0.620i)3-s − 0.144·4-s + 0.495i·5-s + (0.725 + 0.573i)6-s − 1.69i·7-s − 1.05·8-s + (0.230 + 0.973i)9-s + 0.457i·10-s + (0.513 − 0.858i)11-s + (−0.112 − 0.0893i)12-s + 0.668i·13-s − 1.56i·14-s + (−0.307 + 0.388i)15-s − 0.835·16-s − 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.934 - 0.355i$
Analytic conductor: \(1.94706\)
Root analytic conductor: \(1.39537\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :3/2),\ 0.934 - 0.355i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.96887 + 0.361426i\)
\(L(\frac12)\) \(\approx\) \(1.96887 + 0.361426i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.07 - 3.22i)T \)
11 \( 1 + (-18.7 + 31.3i)T \)
good2 \( 1 - 2.61T + 8T^{2} \)
5 \( 1 - 5.53iT - 125T^{2} \)
7 \( 1 + 31.3iT - 343T^{2} \)
13 \( 1 - 31.3iT - 2.19e3T^{2} \)
17 \( 1 + 85.3T + 4.91e3T^{2} \)
19 \( 1 - 67.4iT - 6.85e3T^{2} \)
23 \( 1 - 31.3iT - 1.21e4T^{2} \)
29 \( 1 - 100.T + 2.43e4T^{2} \)
31 \( 1 - 122.T + 2.97e4T^{2} \)
37 \( 1 - 55.8T + 5.06e4T^{2} \)
41 \( 1 + 311.T + 6.89e4T^{2} \)
43 \( 1 + 72.2iT - 7.95e4T^{2} \)
47 \( 1 - 80.3iT - 1.03e5T^{2} \)
53 \( 1 + 614. iT - 1.48e5T^{2} \)
59 \( 1 + 163. iT - 2.05e5T^{2} \)
61 \( 1 + 98.8iT - 2.26e5T^{2} \)
67 \( 1 + 158.T + 3.00e5T^{2} \)
71 \( 1 - 654. iT - 3.57e5T^{2} \)
73 \( 1 - 1.08e3iT - 3.89e5T^{2} \)
79 \( 1 + 986. iT - 4.93e5T^{2} \)
83 \( 1 - 329.T + 5.71e5T^{2} \)
89 \( 1 + 1.19e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.17e3T + 9.12e5T^{2} \)
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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

−16.04762872368555225365877835415, −14.61547899308575596253789267117, −13.96881599869573518268979487893, −13.28811484861372625636079265954, −11.27759624452086825226580466869, −10.03909974218401363979854156348, −8.562396941985450897519292950604, −6.73426905145222999421611740115, −4.48160235690420026896603315875, −3.48191240059266032789049814932, 2.65798632735795558858850491929, 4.77784225500535470199415488227, 6.41729153359563786610934290608, 8.576916951687171039750723379170, 9.252891751793766972696358619269, 12.00408814714542283840079092296, 12.61735483421848254185724693034, 13.59789850048309370382218371916, 15.03227420562284685645272322422, 15.37170729808168272826496504511

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