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Properties

Label 1-185-185.24-r1-0-0
Degree $1$
Conductor $185$
Sign $-0.520 - 0.853i$
Analytic cond. $19.8810$
Root an. cond. $19.8810$
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  + (−0.342 + 0.939i)2-s + (−0.939 + 0.342i)3-s + (−0.766 − 0.642i)4-s − i·6-s + (−0.173 + 0.984i)7-s + (0.866 − 0.5i)8-s + (0.766 − 0.642i)9-s + (0.5 + 0.866i)11-s + (0.939 + 0.342i)12-s + (−0.642 + 0.766i)13-s + (−0.866 − 0.5i)14-s + (0.173 + 0.984i)16-s + (0.642 + 0.766i)17-s + (0.342 + 0.939i)18-s + (0.342 + 0.939i)19-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)2-s + (−0.939 + 0.342i)3-s + (−0.766 − 0.642i)4-s − i·6-s + (−0.173 + 0.984i)7-s + (0.866 − 0.5i)8-s + (0.766 − 0.642i)9-s + (0.5 + 0.866i)11-s + (0.939 + 0.342i)12-s + (−0.642 + 0.766i)13-s + (−0.866 − 0.5i)14-s + (0.173 + 0.984i)16-s + (0.642 + 0.766i)17-s + (0.342 + 0.939i)18-s + (0.342 + 0.939i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.520 - 0.853i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ -0.520 - 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2620195864 + 0.4665274003i\)
\(L(\frac12)\) \(\approx\) \(-0.2620195864 + 0.4665274003i\)
\(L(1)\) \(\approx\) \(0.3920196300 + 0.4321484885i\)
\(L(1)\) \(\approx\) \(0.3920196300 + 0.4321484885i\)

Euler product

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

−26.86720065108749246525816097451, −25.51704742608294301044208950600, −24.203501393131033848124605839008, −23.29295091836760353124439810531, −22.33028236432513354456308600612, −21.75731098682377276600553869448, −20.38595468816026904034439758013, −19.59471256754986012755206316677, −18.629257041040129667784285466863, −17.59722740131484213826469597901, −16.96377883651317630374756612962, −16.013743889644172391463552108140, −14.02005631768681316070237515549, −13.30149991658836644793192401119, −12.17065520677114437023825913505, −11.36556761886941962800784141546, −10.43847774096343232976893455403, −9.579687626179494676411504117389, −7.97086915362320672773614436218, −7.03965249014291159298421370733, −5.486557563722066057667853868186, −4.29309142401398940085797130477, −2.95272389851005254151175629499, −1.15716253194635113089029178073, −0.28891766325173419722850898748, 1.62748180516959586608390583470, 4.05348983566952359228544372604, 5.13687363705244526959703645911, 6.10791718032306443810750635212, 6.96830082768764242191599417343, 8.38208878831496372003416855177, 9.62920576978129674116824076734, 10.20986885255837860466800976686, 11.91210542987856300407298369651, 12.517896801249825867663276282915, 14.285610939463231968450435704785, 15.07945145129576602020685607573, 16.06276246781071005067004756378, 16.83119478568878450291863351139, 17.75613772170127928695788076500, 18.571103677962983684120347971369, 19.57191873442353863400355684661, 21.25495987931925056818498390360, 22.16280532308697080815886304547, 22.8841261450152659570003614793, 23.84486610612456280301395996169, 24.76508948396322510457035284722, 25.63744685952832408342097972226, 26.74170414378143305539162395777, 27.58381952523828223298546532312

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