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Properties

Label 2-183-61.60-c1-0-1
Degree $2$
Conductor $183$
Sign $-0.640 - 0.768i$
Analytic cond. $1.46126$
Root an. cond. $1.20882$
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  + i·2-s − 3-s + 4-s − 3·5-s − i·6-s + 3i·7-s + 3i·8-s + 9-s − 3i·10-s + 5i·11-s − 12-s − 3·13-s − 3·14-s + 3·15-s − 16-s − 4i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s + 0.5·4-s − 1.34·5-s − 0.408i·6-s + 1.13i·7-s + 1.06i·8-s + 0.333·9-s − 0.948i·10-s + 1.50i·11-s − 0.288·12-s − 0.832·13-s − 0.801·14-s + 0.774·15-s − 0.250·16-s − 0.970i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.640 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(183\)    =    \(3 \cdot 61\)
Sign: $-0.640 - 0.768i$
Analytic conductor: \(1.46126\)
Root analytic conductor: \(1.20882\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{183} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 183,\ (\ :1/2),\ -0.640 - 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.352886 + 0.753426i\)
\(L(\frac12)\) \(\approx\) \(0.352886 + 0.753426i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
61 \( 1 + (-5 - 6i)T \)
good2 \( 1 - iT - 2T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 - 5iT - 11T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 5iT - 23T^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 5iT - 59T^{2} \)
67 \( 1 - 9iT - 67T^{2} \)
71 \( 1 - 8iT - 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + 3iT - 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 8iT - 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−12.42474012211004117362969939208, −12.03275366043350645566575763178, −11.39869315293051431734542700439, −10.03035317727548579123937049021, −8.732226370206352620483405693586, −7.38501406205036986178529194257, −7.12934477315536004511672067531, −5.53808300380591561298358756678, −4.63390078721595918380726259410, −2.59830876304975568371731392798, 0.824228509759666030386987914820, 3.30582233141936790846060213484, 4.15372595930918359834753591262, 5.92356202959149299113874084056, 7.30599296661241124529063956230, 7.84156626296526906831417447007, 9.634479739525449549224183348266, 10.76895461471631487340119504408, 11.27245096947511019225311191120, 11.93858844716338639612920632242

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