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Properties

Label 2-183-183.50-c1-0-5
Degree $2$
Conductor $183$
Sign $-0.365 + 0.930i$
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  + (−1.53 − 1.53i)2-s + (−1.73 − 0.0678i)3-s + 2.73i·4-s + 3.09·5-s + (2.55 + 2.76i)6-s + (1.25 − 1.25i)7-s + (1.13 − 1.13i)8-s + (2.99 + 0.234i)9-s + (−4.76 − 4.76i)10-s + (2.26 − 2.26i)11-s + (0.185 − 4.73i)12-s − 4.21·13-s − 3.85·14-s + (−5.36 − 0.210i)15-s + 1.98·16-s + (−1.04 + 1.04i)17-s + ⋯
L(s)  = 1  + (−1.08 − 1.08i)2-s + (−0.999 − 0.0391i)3-s + 1.36i·4-s + 1.38·5-s + (1.04 + 1.13i)6-s + (0.473 − 0.473i)7-s + (0.401 − 0.401i)8-s + (0.996 + 0.0782i)9-s + (−1.50 − 1.50i)10-s + (0.682 − 0.682i)11-s + (0.0535 − 1.36i)12-s − 1.16·13-s − 1.03·14-s + (−1.38 − 0.0542i)15-s + 0.495·16-s + (−0.254 + 0.254i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(183\)    =    \(3 \cdot 61\)
Sign: $-0.365 + 0.930i$
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: \((2,\ 183,\ (\ :1/2),\ -0.365 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.355481 - 0.521544i\)
\(L(\frac12)\) \(\approx\) \(0.355481 - 0.521544i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0678i)T \)
61 \( 1 + (5.45 - 5.58i)T \)
good2 \( 1 + (1.53 + 1.53i)T + 2iT^{2} \)
5 \( 1 - 3.09T + 5T^{2} \)
7 \( 1 + (-1.25 + 1.25i)T - 7iT^{2} \)
11 \( 1 + (-2.26 + 2.26i)T - 11iT^{2} \)
13 \( 1 + 4.21T + 13T^{2} \)
17 \( 1 + (1.04 - 1.04i)T - 17iT^{2} \)
19 \( 1 + 7.74iT - 19T^{2} \)
23 \( 1 + (-1.17 - 1.17i)T + 23iT^{2} \)
29 \( 1 + (-5.76 + 5.76i)T - 29iT^{2} \)
31 \( 1 + (0.319 + 0.319i)T + 31iT^{2} \)
37 \( 1 + (-6.69 - 6.69i)T + 37iT^{2} \)
41 \( 1 + 1.90T + 41T^{2} \)
43 \( 1 + (2.06 + 2.06i)T + 43iT^{2} \)
47 \( 1 + 8.20iT - 47T^{2} \)
53 \( 1 + (-5.28 - 5.28i)T + 53iT^{2} \)
59 \( 1 + (0.557 - 0.557i)T - 59iT^{2} \)
67 \( 1 + (-1.48 - 1.48i)T + 67iT^{2} \)
71 \( 1 + (10.0 - 10.0i)T - 71iT^{2} \)
73 \( 1 - 4.20T + 73T^{2} \)
79 \( 1 + (8.87 - 8.87i)T - 79iT^{2} \)
83 \( 1 - 1.37iT - 83T^{2} \)
89 \( 1 + (-1.44 + 1.44i)T - 89iT^{2} \)
97 \( 1 + 14.1iT - 97T^{2} \)
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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

−11.82661919051994100741523254828, −11.25445558569429306700142897785, −10.26481908651114085221039548244, −9.726914845645851527050348017142, −8.727352550086203618795454265010, −7.17687799190018480043258460329, −5.99632236311414513143589274590, −4.68607085755413356559318441077, −2.46255828008342419470814608990, −1.02584095397114791470542186209, 1.68992835723828263744370990370, 4.90186482422356482840783455385, 5.86807235736085727758038242463, 6.63166938366216782217324499445, 7.66286401849529282433041364451, 9.080479372466951267674689007713, 9.817140043289289608940352711050, 10.44168296671530240122313043654, 11.98296159098768468315158966440, 12.72766888259341671863437821045

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