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Properties

Label 1-187-187.14-r1-0-0
Degree $1$
Conductor $187$
Sign $0.0720 - 0.997i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0720 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0720 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.0720 - 0.997i$
Analytic conductor: \(20.0959\)
Root analytic conductor: \(20.0959\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (1:\ ),\ 0.0720 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.908151439 - 1.775236663i\)
\(L(\frac12)\) \(\approx\) \(1.908151439 - 1.775236663i\)
\(L(1)\) \(\approx\) \(1.293866627 - 0.7385429527i\)
\(L(1)\) \(\approx\) \(1.293866627 - 0.7385429527i\)

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

−26.901528121358850990480039833067, −25.91647525025268496553241207042, −25.36007594939590410324912626124, −24.7323714694014408958808707938, −23.75863993437851907973765144236, −22.2259472166348673285309366158, −21.48523126903789830935626154695, −20.39845747533982547338310754723, −19.26527042916710753009437955389, −18.24724082931407990086899305697, −17.68853922511793440825997968209, −16.185873073407005503159953264990, −15.51559760024355754182490760407, −14.46886750669367209351066913573, −13.755885599930401616459052185241, −12.75481311958812609881826219255, −10.75518336235433862367571031372, −9.73108817690843960349660444418, −8.82901980420017273237961777693, −8.26375482774346699593621234903, −6.74240758628521145851927408372, −5.68955164542991570487422075348, −4.55886345465518419242497557550, −2.70441327630318203407416291952, −1.43186208753651851849272014911, 1.148324049202517442322142842396, 2.0298472721589174590138382694, 3.37739986802255863796913139601, 4.391672484442671448554608162757, 6.386133931255826255806750291456, 7.761246057404207638728282771162, 8.611130731499178244286965795751, 9.83090012877591691413339875062, 10.334270274710633234649098839235, 11.7443805230561324275824943792, 13.15836332465291712136275294615, 13.65202792990539418052740620301, 14.45731284432818109869288554443, 16.21029282177450667166294947692, 17.33224885756402557171687378880, 18.19645639754128761339495285920, 19.030892774639573396787419888388, 20.12772730852485824125508405538, 20.83956557257343665644353792691, 21.39762599250714416913708873885, 22.706109398319131320565757978277, 23.905780497708103409784779743703, 25.217407530112996838431488659253, 25.91520080874024200206368040860, 26.609197253792883219634855575779

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