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Properties

Label 1-187-187.3-r1-0-0
Degree $1$
Conductor $187$
Sign $-0.925 + 0.377i$
Analytic cond. $20.0959$
Root an. cond. $20.0959$
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  + (−0.453 − 0.891i)2-s + (−0.972 + 0.233i)3-s + (−0.587 + 0.809i)4-s + (−0.996 − 0.0784i)5-s + (0.649 + 0.760i)6-s + (−0.233 + 0.972i)7-s + (0.987 + 0.156i)8-s + (0.891 − 0.453i)9-s + (0.382 + 0.923i)10-s + (0.382 − 0.923i)12-s + (0.951 + 0.309i)13-s + (0.972 − 0.233i)14-s + (0.987 − 0.156i)15-s + (−0.309 − 0.951i)16-s + (−0.809 − 0.587i)18-s + (−0.156 + 0.987i)19-s + ⋯
L(s)  = 1  + (−0.453 − 0.891i)2-s + (−0.972 + 0.233i)3-s + (−0.587 + 0.809i)4-s + (−0.996 − 0.0784i)5-s + (0.649 + 0.760i)6-s + (−0.233 + 0.972i)7-s + (0.987 + 0.156i)8-s + (0.891 − 0.453i)9-s + (0.382 + 0.923i)10-s + (0.382 − 0.923i)12-s + (0.951 + 0.309i)13-s + (0.972 − 0.233i)14-s + (0.987 − 0.156i)15-s + (−0.309 − 0.951i)16-s + (−0.809 − 0.587i)18-s + (−0.156 + 0.987i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.925 + 0.377i$
Analytic conductor: \(20.0959\)
Root analytic conductor: \(20.0959\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (1:\ ),\ -0.925 + 0.377i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02591354781 + 0.1321341185i\)
\(L(\frac12)\) \(\approx\) \(0.02591354781 + 0.1321341185i\)
\(L(1)\) \(\approx\) \(0.4527695799 - 0.03734925601i\)
\(L(1)\) \(\approx\) \(0.4527695799 - 0.03734925601i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.453 - 0.891i)T \)
3 \( 1 + (-0.972 + 0.233i)T \)
5 \( 1 + (-0.996 - 0.0784i)T \)
7 \( 1 + (-0.233 + 0.972i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (-0.156 + 0.987i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + (-0.852 + 0.522i)T \)
31 \( 1 + (-0.649 + 0.760i)T \)
37 \( 1 + (-0.522 - 0.852i)T \)
41 \( 1 + (0.852 + 0.522i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (-0.587 - 0.809i)T \)
53 \( 1 + (-0.453 - 0.891i)T \)
59 \( 1 + (0.156 + 0.987i)T \)
61 \( 1 + (-0.760 + 0.649i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.0784 + 0.996i)T \)
73 \( 1 + (0.852 - 0.522i)T \)
79 \( 1 + (0.0784 + 0.996i)T \)
83 \( 1 + (-0.891 - 0.453i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.760 - 0.649i)T \)
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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

−26.57303256572819824604257979296, −25.6939884776028467482755137378, −24.277765333905795937388152422534, −23.72194911284745539315182466733, −22.97694915908417508445003993494, −22.35718039379650078374983717790, −20.56886262654201352545311380507, −19.37588115153244095358450889894, −18.68400253017113170760196010719, −17.54722766083564107145141532044, −16.81355093820365091834163868909, −15.907550536527349554453612921817, −15.205661845154839597857610435326, −13.66810231582066326295008449301, −12.81814494475560777846179760174, −11.166923907284350474656295426220, −10.74361200141901917148510881041, −9.2914938286506318703485014690, −7.84524902999167756460662039074, −7.158404951401852977799186479, −6.20116934861415828098103422963, −4.88043682379689996064231512652, −3.84330900154979006768147335994, −1.067960389073044933983747191151, −0.08376205550718522805963515083, 1.43474874540935648656805518082, 3.27832383041626896878647999281, 4.29161737316933014024569422166, 5.57981485758223145261818725649, 7.067357236895087240902553072403, 8.43995452820795100696382533310, 9.33776543244975782853701148838, 10.72400767875190659046500987868, 11.356318874659261340273875955666, 12.30154169699214001110993304718, 12.89509618482961574227302710680, 14.78400969799788066832623740037, 16.08376088645298861628786681507, 16.55041457885988577611983447575, 18.01474779936514253131220876314, 18.63194121547652471688900124782, 19.46488366949909462449969619189, 20.81450242138472174972976367102, 21.46502561735940898801341195660, 22.69379997860939627162233657935, 23.04297674503862680389641643738, 24.36859806844786438609052623273, 25.70887808210266890331059875043, 26.8502307706059633580575666721, 27.61578546864454510895559970991

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