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Properties

Label 2-183-183.50-c1-0-4
Degree $2$
Conductor $183$
Sign $0.465 + 0.884i$
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  + (−1.79 − 1.79i)2-s + (1.70 + 0.303i)3-s + 4.47i·4-s + 0.391·5-s + (−2.52 − 3.61i)6-s + (−0.663 + 0.663i)7-s + (4.46 − 4.46i)8-s + (2.81 + 1.03i)9-s + (−0.704 − 0.704i)10-s + (2.18 − 2.18i)11-s + (−1.35 + 7.63i)12-s + 5.70·13-s + 2.38·14-s + (0.667 + 0.118i)15-s − 7.10·16-s + (−2.44 + 2.44i)17-s + ⋯
L(s)  = 1  + (−1.27 − 1.27i)2-s + (0.984 + 0.175i)3-s + 2.23i·4-s + 0.175·5-s + (−1.02 − 1.47i)6-s + (−0.250 + 0.250i)7-s + (1.57 − 1.57i)8-s + (0.938 + 0.345i)9-s + (−0.222 − 0.222i)10-s + (0.659 − 0.659i)11-s + (−0.392 + 2.20i)12-s + 1.58·13-s + 0.638·14-s + (0.172 + 0.0306i)15-s − 1.77·16-s + (−0.593 + 0.593i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.465 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(183\)    =    \(3 \cdot 61\)
Sign: $0.465 + 0.884i$
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: \((2,\ 183,\ (\ :1/2),\ 0.465 + 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.780744 - 0.471343i\)
\(L(\frac12)\) \(\approx\) \(0.780744 - 0.471343i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 - 0.303i)T \)
61 \( 1 + (1.72 + 7.61i)T \)
good2 \( 1 + (1.79 + 1.79i)T + 2iT^{2} \)
5 \( 1 - 0.391T + 5T^{2} \)
7 \( 1 + (0.663 - 0.663i)T - 7iT^{2} \)
11 \( 1 + (-2.18 + 2.18i)T - 11iT^{2} \)
13 \( 1 - 5.70T + 13T^{2} \)
17 \( 1 + (2.44 - 2.44i)T - 17iT^{2} \)
19 \( 1 + 4.76iT - 19T^{2} \)
23 \( 1 + (2.39 + 2.39i)T + 23iT^{2} \)
29 \( 1 + (2.95 - 2.95i)T - 29iT^{2} \)
31 \( 1 + (-6.42 - 6.42i)T + 31iT^{2} \)
37 \( 1 + (2.88 + 2.88i)T + 37iT^{2} \)
41 \( 1 - 3.26T + 41T^{2} \)
43 \( 1 + (7.40 + 7.40i)T + 43iT^{2} \)
47 \( 1 - 10.0iT - 47T^{2} \)
53 \( 1 + (1.16 + 1.16i)T + 53iT^{2} \)
59 \( 1 + (5.04 - 5.04i)T - 59iT^{2} \)
67 \( 1 + (2.77 + 2.77i)T + 67iT^{2} \)
71 \( 1 + (-8.79 + 8.79i)T - 71iT^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 + (-1.86 + 1.86i)T - 79iT^{2} \)
83 \( 1 + 5.41iT - 83T^{2} \)
89 \( 1 + (10.3 - 10.3i)T - 89iT^{2} \)
97 \( 1 - 3.49iT - 97T^{2} \)
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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

−12.27029491470699474525568541399, −11.09887757892111977017105038648, −10.48191211623022350676561979551, −9.239507467570126724689672393107, −8.833261558288272100472496220942, −8.020849651035654325151885749790, −6.49337371986545808806528290582, −3.99610854106438828269914687889, −3.01011858064585466012841349551, −1.57755584116678471696591398618, 1.60206882528578268639274470653, 3.98571492973640969983677586518, 5.99165430035742500317840025003, 6.82035127241513251160205596109, 7.87355685664375100125666719110, 8.569312442977295809354591970117, 9.609392277315516036725921778663, 10.06300429299037426641828884041, 11.61637128689975638594586949195, 13.33306067701284422007225683491

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