↑

Properties

Label 1-187-187.7-r0-0-0
Degree $1$
Conductor $187$
Sign $-0.217 - 0.976i$
Analytic cond. $0.868424$
Root an. cond. $0.868424$
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  + (−0.453 + 0.891i)2-s + (−0.233 + 0.972i)3-s + (−0.587 − 0.809i)4-s + (0.0784 + 0.996i)5-s + (−0.760 − 0.649i)6-s + (−0.972 + 0.233i)7-s + (0.987 − 0.156i)8-s + (−0.891 − 0.453i)9-s + (−0.923 − 0.382i)10-s + (0.923 − 0.382i)12-s + (−0.951 + 0.309i)13-s + (0.233 − 0.972i)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 + (−0.233 + 0.972i)3-s + (−0.587 − 0.809i)4-s + (0.0784 + 0.996i)5-s + (−0.760 − 0.649i)6-s + (−0.972 + 0.233i)7-s + (0.987 − 0.156i)8-s + (−0.891 − 0.453i)9-s + (−0.923 − 0.382i)10-s + (0.923 − 0.382i)12-s + (−0.951 + 0.309i)13-s + (0.233 − 0.972i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.217 - 0.976i$
Analytic conductor: \(0.868424\)
Root analytic conductor: \(0.868424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (0:\ ),\ -0.217 - 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1782268108 + 0.2222909942i\)
\(L(\frac12)\) \(\approx\) \(-0.1782268108 + 0.2222909942i\)
\(L(1)\) \(\approx\) \(0.3023564902 + 0.4214980623i\)
\(L(1)\) \(\approx\) \(0.3023564902 + 0.4214980623i\)

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 \( 1 + (-0.233 + 0.972i)T \)
5 \( 1 + (0.0784 + 0.996i)T \)
7 \( 1 + (-0.972 + 0.233i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (-0.156 - 0.987i)T \)
23 \( 1 + (-0.382 + 0.923i)T \)
29 \( 1 + (0.522 - 0.852i)T \)
31 \( 1 + (-0.760 + 0.649i)T \)
37 \( 1 + (0.852 + 0.522i)T \)
41 \( 1 + (-0.522 - 0.852i)T \)
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 \( 1 + (-0.649 + 0.760i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.996 + 0.0784i)T \)
73 \( 1 + (-0.522 + 0.852i)T \)
79 \( 1 + (-0.996 - 0.0784i)T \)
83 \( 1 + (-0.891 + 0.453i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.649 + 0.760i)T \)
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

−26.60280329740430878310756464477, −25.45542849865224378785548217446, −24.75581343178498125037363844198, −23.516669735449174277119684896424, −22.64317752208930210595288182767, −21.67519523534265991636754464460, −20.23956049474759804030072232673, −19.86616935986504980657704961576, −18.8717366406874324443707101342, −17.93948693891881934799373100641, −16.825807419558575248511591209942, −16.41994383240063916835613502441, −14.29414021245445206418732220356, −13.05930753500083527890510458185, −12.63624837488558229422535059484, −11.85662950461131378437650245700, −10.434926456261390208106920955292, −9.426650147869577707066437740499, −8.32623971548774409862503228249, −7.37004999305414992936386825787, −5.911235958747992388969640483170, −4.49266183915079805180741260431, −2.95052396168730437437666603115, −1.663101044301576023690837538004, −0.26130719454353206405083466449, 2.70644457391907623293248773717, 4.106341961921620899327851883311, 5.45714417536719996955830828143, 6.4247391828819365921800948033, 7.3727574200934668304050383540, 8.978884285820589712466283293404, 9.77543440623926883893737036994, 10.49383338057338225302066767519, 11.73993698768502964444458191070, 13.480149255771499890602685886674, 14.551803701200446700735914772234, 15.34444505609921736025059159447, 16.04237022037596193048455616004, 17.121767354884701166836017363471, 17.91307413746481383570152881376, 19.18083953133387189833591686068, 19.796228938081267194247698740338, 21.622048956183990641322221852241, 22.23324817465658584058493138984, 22.983804782009045115615833249110, 23.99988351169132258762745762646, 25.5441179434563152856230274809, 25.889540082714002087048892643052, 26.84271724563980203820495320441, 27.45423519806550844813262620662

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