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Properties

Label 1-185-185.2-r0-0-0
Degree $1$
Conductor $185$
Sign $-0.383 + 0.923i$
Analytic cond. $0.859136$
Root an. cond. $0.859136$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.984 + 0.173i)3-s + (−0.939 − 0.342i)4-s − i·6-s + (0.642 + 0.766i)7-s + (0.5 − 0.866i)8-s + (0.939 − 0.342i)9-s + (0.5 − 0.866i)11-s + (0.984 + 0.173i)12-s + (0.939 + 0.342i)13-s + (−0.866 + 0.5i)14-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + (0.173 + 0.984i)18-s + (−0.984 + 0.173i)19-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.984 + 0.173i)3-s + (−0.939 − 0.342i)4-s − i·6-s + (0.642 + 0.766i)7-s + (0.5 − 0.866i)8-s + (0.939 − 0.342i)9-s + (0.5 − 0.866i)11-s + (0.984 + 0.173i)12-s + (0.939 + 0.342i)13-s + (−0.866 + 0.5i)14-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + (0.173 + 0.984i)18-s + (−0.984 + 0.173i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.383 + 0.923i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ -0.383 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4133419488 + 0.6191032548i\)
\(L(\frac12)\) \(\approx\) \(0.4133419488 + 0.6191032548i\)
\(L(1)\) \(\approx\) \(0.6095006632 + 0.4307870007i\)
\(L(1)\) \(\approx\) \(0.6095006632 + 0.4307870007i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (-0.984 + 0.173i)T \)
7 \( 1 + (0.642 + 0.766i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.939 + 0.342i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + iT \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 - T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.642 - 0.766i)T \)
59 \( 1 + (-0.642 + 0.766i)T \)
61 \( 1 + (0.342 - 0.939i)T \)
67 \( 1 + (-0.642 - 0.766i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.642 + 0.766i)T \)
83 \( 1 + (0.342 + 0.939i)T \)
89 \( 1 + (0.642 - 0.766i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.29367218280988739584147158220, −26.33650890543952801185775601284, −24.939129613984658687934943504092, −23.63451594071355399271387852205, −22.98763170497332526107988859167, −22.19906116462534912753369422982, −21.03325313260316601483881400616, −20.329872319779063246961408925469, −19.17783929590471195420828350382, −18.04099593961899212695698317376, −17.500196401994100688023773176711, −16.641997522506763001451255673494, −15.11060322598274077637395101675, −13.69132854727224037789541654223, −12.873645025295302281617483736617, −11.81390410631789842323894708037, −10.92348384081091752660612839974, −10.317587822374784622707567734154, −8.91743628498037935165767225381, −7.61514910352789132112352396405, −6.35463362893867482552614407216, −4.715286715982458638624444192757, −4.14082336401093184882983427268, −2.129070754174229617028140193000, −0.85847799032040649362385737024, 1.357709140542073035579038585241, 3.91798230244795901354201897585, 5.04671187538029252373963250624, 6.03746240283273018987788604351, 6.76666957601590843376269784763, 8.38824980115745691259935972533, 9.07532457999842933607610387709, 10.609792263769073507714760921678, 11.45921238460964301530618393524, 12.7499709872938350992187173264, 13.90057165572713596776730236869, 15.10924395503537756553739736623, 15.870776723335953756976572616826, 16.797879224400893719173193083501, 17.69809947820272989270257573718, 18.42330044741402859241362099570, 19.40476843840613840106739908736, 21.447582120813315307024089850561, 21.72152421523640081832816254822, 23.02607277581984863606194005085, 23.74323863610563521047328962346, 24.56085557688259634095510877796, 25.45305063967400297942201160318, 26.7396384346959051216266200275, 27.45864977047522917058102353320

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