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Properties

Label 2-183-183.50-c1-0-11
Degree $2$
Conductor $183$
Sign $0.968 + 0.248i$
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  + (0.977 + 0.977i)2-s + (−1.38 − 1.03i)3-s − 0.0900i·4-s + 0.568·5-s + (−0.343 − 2.36i)6-s + (1.94 − 1.94i)7-s + (2.04 − 2.04i)8-s + (0.852 + 2.87i)9-s + (0.555 + 0.555i)10-s + (0.422 − 0.422i)11-s + (−0.0933 + 0.125i)12-s − 1.07·13-s + 3.79·14-s + (−0.788 − 0.588i)15-s + 3.81·16-s + (−0.499 + 0.499i)17-s + ⋯
L(s)  = 1  + (0.691 + 0.691i)2-s + (−0.801 − 0.598i)3-s − 0.0450i·4-s + 0.254·5-s + (−0.140 − 0.967i)6-s + (0.734 − 0.734i)7-s + (0.722 − 0.722i)8-s + (0.284 + 0.958i)9-s + (0.175 + 0.175i)10-s + (0.127 − 0.127i)11-s + (−0.0269 + 0.0360i)12-s − 0.298·13-s + 1.01·14-s + (−0.203 − 0.152i)15-s + 0.952·16-s + (−0.121 + 0.121i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(183\)    =    \(3 \cdot 61\)
Sign: $0.968 + 0.248i$
Analytic conductor: \(1.46126\)
Root analytic conductor: \(1.20882\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{183} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 183,\ (\ :1/2),\ 0.968 + 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42729 - 0.180506i\)
\(L(\frac12)\) \(\approx\) \(1.42729 - 0.180506i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.38 + 1.03i)T \)
61 \( 1 + (-7.80 - 0.377i)T \)
good2 \( 1 + (-0.977 - 0.977i)T + 2iT^{2} \)
5 \( 1 - 0.568T + 5T^{2} \)
7 \( 1 + (-1.94 + 1.94i)T - 7iT^{2} \)
11 \( 1 + (-0.422 + 0.422i)T - 11iT^{2} \)
13 \( 1 + 1.07T + 13T^{2} \)
17 \( 1 + (0.499 - 0.499i)T - 17iT^{2} \)
19 \( 1 - 3.23iT - 19T^{2} \)
23 \( 1 + (0.169 + 0.169i)T + 23iT^{2} \)
29 \( 1 + (4.76 - 4.76i)T - 29iT^{2} \)
31 \( 1 + (0.431 + 0.431i)T + 31iT^{2} \)
37 \( 1 + (-7.29 - 7.29i)T + 37iT^{2} \)
41 \( 1 + 7.90T + 41T^{2} \)
43 \( 1 + (-6.00 - 6.00i)T + 43iT^{2} \)
47 \( 1 + 5.26iT - 47T^{2} \)
53 \( 1 + (-2.34 - 2.34i)T + 53iT^{2} \)
59 \( 1 + (1.32 - 1.32i)T - 59iT^{2} \)
67 \( 1 + (0.316 + 0.316i)T + 67iT^{2} \)
71 \( 1 + (-4.85 + 4.85i)T - 71iT^{2} \)
73 \( 1 + 15.4T + 73T^{2} \)
79 \( 1 + (-2.07 + 2.07i)T - 79iT^{2} \)
83 \( 1 - 14.5iT - 83T^{2} \)
89 \( 1 + (-4.31 + 4.31i)T - 89iT^{2} \)
97 \( 1 - 2.35iT - 97T^{2} \)
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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

−12.85941600699928291261081897493, −11.68628626850978816352386151323, −10.74391203993616006076584012204, −9.869706767629959632743030218396, −7.998038730593182288337701425797, −7.17438671799026419226521023186, −6.16298002025768534913998660927, −5.25865058694492081507126571946, −4.20991661326067442282851741305, −1.48923062865039371259906934409, 2.24888316207926219698935033267, 3.93090831550451928270420364917, 4.95689829378958583839361560244, 5.84225101581541951357888127056, 7.46949910246775552880881791650, 8.856792485798211834328604760003, 9.952231556499169890315993926211, 11.16055961990696824220469252654, 11.59845577771357247481238277974, 12.44021137737955483795673752014

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