Properties

Label 1-185-185.54-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: $\chi_{185} (54, \cdot )$
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

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

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