↑

Properties

Label 1-185-185.34-r0-0-0
Degree $1$
Conductor $185$
Sign $-0.108 - 0.994i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.108 - 0.994i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ -0.108 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4714116426 - 0.5255531505i\)
\(L(\frac12)\) \(\approx\) \(0.4714116426 - 0.5255531505i\)
\(L(1)\) \(\approx\) \(0.6910774477 - 0.2942810393i\)
\(L(1)\) \(\approx\) \(0.6910774477 - 0.2942810393i\)

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

Imaginary part of the first few zeros on the critical line

−27.51369123015345070281096702823, −26.000065189233800303265249155121, −25.47539064343747573667062787302, −24.85725747120173179263829338953, −23.51829489728744933534206721760, −23.15470324858096793592922764912, −22.18982608124203452990385813233, −20.67932306008015535707270196843, −19.208223644553342093467540583, −18.635001796672491767053075969944, −17.87426203308037342375720601926, −16.7378164566998892740624826546, −15.88212295418543904010833288320, −14.78048681362335414099739355653, −13.6615433436446073223290598772, −12.84295574065719229866400148402, −11.89509369724949712165306215163, −10.196736206283810165066178817757, −9.072229025495961045029127367895, −7.98679120324975848000092294654, −7.04341338186757891411883020901, −6.0649869472189891440304213317, −5.20289459927054009925441803907, −3.32158488512919809759480363195, −1.48742627191480763831722546848, 0.66574933394857857115442934708, 2.95927892364641782592358106466, 3.63229979407025853945438770539, 4.878807160265732303039139309602, 6.14474173669930604676486621997, 8.08780781617658739824283358599, 9.07958017600040453141118402049, 10.18780705601562627845615212688, 10.73960091822641900739990454375, 11.75972768173266197462326170529, 13.14737285555599534477507174959, 13.85411558444783373521725800042, 15.30191748271605709540742433136, 16.44684997215694192342567252117, 17.12599689566357614482709891021, 18.451425694768897156753803846283, 19.39807523873859652234982419267, 20.43103815248067303810758621341, 21.08195559862151460670855863923, 22.01447252772169974783398496223, 22.90991740538662762913094947553, 23.59298556895192759702288072108, 25.5231654233437332177010849581, 26.47479947757998014657375215974, 26.853033491663088887421755893160

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