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Properties

Label 2-183-61.34-c1-0-2
Degree $2$
Conductor $183$
Sign $-0.218 - 0.975i$
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  + (−2.22 + 1.61i)2-s + (−0.809 + 0.587i)3-s + (1.71 − 5.26i)4-s + (0.147 − 0.454i)5-s + (0.848 − 2.61i)6-s + (1.12 + 0.818i)7-s + (2.99 + 9.22i)8-s + (0.309 − 0.951i)9-s + (0.405 + 1.24i)10-s + 1.98·11-s + (1.71 + 5.26i)12-s + 4.05·13-s − 3.82·14-s + (0.147 + 0.454i)15-s + (−12.5 − 9.14i)16-s + (−1.72 + 5.30i)17-s + ⋯
L(s)  = 1  + (−1.57 + 1.14i)2-s + (−0.467 + 0.339i)3-s + (0.855 − 2.63i)4-s + (0.0660 − 0.203i)5-s + (0.346 − 1.06i)6-s + (0.426 + 0.309i)7-s + (1.05 + 3.26i)8-s + (0.103 − 0.317i)9-s + (0.128 + 0.394i)10-s + 0.598·11-s + (0.493 + 1.51i)12-s + 1.12·13-s − 1.02·14-s + (0.0381 + 0.117i)15-s + (−3.14 − 2.28i)16-s + (−0.418 + 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(183\)    =    \(3 \cdot 61\)
Sign: $-0.218 - 0.975i$
Analytic conductor: \(1.46126\)
Root analytic conductor: \(1.20882\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{183} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 183,\ (\ :1/2),\ -0.218 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.327192 + 0.408609i\)
\(L(\frac12)\) \(\approx\) \(0.327192 + 0.408609i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (1.03 + 7.74i)T \)
good2 \( 1 + (2.22 - 1.61i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-0.147 + 0.454i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.12 - 0.818i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 - 1.98T + 11T^{2} \)
13 \( 1 - 4.05T + 13T^{2} \)
17 \( 1 + (1.72 - 5.30i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.58 - 4.05i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.47 + 4.55i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 - 4.19T + 29T^{2} \)
31 \( 1 + (-1.28 - 0.936i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-6.33 - 4.60i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.16 - 3.02i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-3.55 - 10.9i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + 6.57T + 47T^{2} \)
53 \( 1 + (2.95 + 9.09i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.09 + 1.52i)T + (18.2 - 56.1i)T^{2} \)
67 \( 1 + (0.341 - 1.05i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-1.39 - 4.29i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.13 + 3.48i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.74 + 11.5i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (6.64 - 4.82i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-12.7 + 9.24i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (7.61 + 5.53i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.88392644175946616998631365087, −11.30685742027431149768548244352, −10.71496450834950183747653801360, −9.739516534529261811053118372531, −8.563155173603253181438521222357, −8.261752872602507144922136821840, −6.47529287583899880963358483588, −6.14247877248584557500515980568, −4.63464547183717579796370586704, −1.45722537728540853926722963808, 0.982208444974824225107962005408, 2.56703083091743149660595730429, 4.22322395758445688134513907070, 6.55526201607133266735683760368, 7.43348371144075030371139722566, 8.615254949928951218220861789047, 9.329875693529197883628808083365, 10.67666172424503353120455270186, 11.08849829544862388966690800108, 11.88271038539140252338747849236

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