Properties

Label 1-191-191.28-r1-0-0
Degree $1$
Conductor $191$
Sign $0.644 + 0.764i$
Analytic cond. $20.5258$
Root an. cond. $20.5258$
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  + (0.991 − 0.131i)2-s + (0.701 + 0.712i)3-s + (0.965 − 0.261i)4-s + (0.546 + 0.837i)5-s + (0.789 + 0.614i)6-s + (0.809 + 0.587i)7-s + (0.922 − 0.386i)8-s + (−0.0165 + 0.999i)9-s + (0.652 + 0.757i)10-s + (0.677 − 0.735i)11-s + (0.863 + 0.504i)12-s + (−0.461 − 0.887i)13-s + (0.879 + 0.475i)14-s + (−0.213 + 0.976i)15-s + (0.863 − 0.504i)16-s + (−0.846 − 0.533i)17-s + ⋯
L(s)  = 1  + (0.991 − 0.131i)2-s + (0.701 + 0.712i)3-s + (0.965 − 0.261i)4-s + (0.546 + 0.837i)5-s + (0.789 + 0.614i)6-s + (0.809 + 0.587i)7-s + (0.922 − 0.386i)8-s + (−0.0165 + 0.999i)9-s + (0.652 + 0.757i)10-s + (0.677 − 0.735i)11-s + (0.863 + 0.504i)12-s + (−0.461 − 0.887i)13-s + (0.879 + 0.475i)14-s + (−0.213 + 0.976i)15-s + (0.863 − 0.504i)16-s + (−0.846 − 0.533i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.644 + 0.764i$
Analytic conductor: \(20.5258\)
Root analytic conductor: \(20.5258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ 0.644 + 0.764i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.826849172 + 2.244225669i\)
\(L(\frac12)\) \(\approx\) \(4.826849172 + 2.244225669i\)
\(L(1)\) \(\approx\) \(2.726809142 + 0.7557284331i\)
\(L(1)\) \(\approx\) \(2.726809142 + 0.7557284331i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.991 - 0.131i)T \)
3 \( 1 + (0.701 + 0.712i)T \)
5 \( 1 + (0.546 + 0.837i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (0.677 - 0.735i)T \)
13 \( 1 + (-0.461 - 0.887i)T \)
17 \( 1 + (-0.846 - 0.533i)T \)
19 \( 1 + (-0.652 + 0.757i)T \)
23 \( 1 + (-0.518 - 0.854i)T \)
29 \( 1 + (0.995 - 0.0990i)T \)
31 \( 1 + (-0.945 - 0.324i)T \)
37 \( 1 + (-0.945 + 0.324i)T \)
41 \( 1 + (0.879 - 0.475i)T \)
43 \( 1 + (0.828 - 0.560i)T \)
47 \( 1 + (-0.490 - 0.871i)T \)
53 \( 1 + (-0.115 + 0.993i)T \)
59 \( 1 + (-0.956 + 0.293i)T \)
61 \( 1 + (-0.371 + 0.928i)T \)
67 \( 1 + (0.997 + 0.0660i)T \)
71 \( 1 + (-0.180 - 0.983i)T \)
73 \( 1 + (0.909 - 0.416i)T \)
79 \( 1 + (-0.995 - 0.0990i)T \)
83 \( 1 + (0.518 - 0.854i)T \)
89 \( 1 + (0.340 + 0.940i)T \)
97 \( 1 + (0.601 + 0.799i)T \)
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

−26.25073092118688693494802907029, −25.537992611188164513730118297069, −24.46267376266027361900959929079, −24.1271945380598196117216907715, −23.24658370672427552294740283861, −21.70263868822521865443356744474, −21.094768138940365182251326289609, −19.9514728091371497867368296575, −19.66294099851841731999172067140, −17.624925808392001063208654064941, −17.23579223049043284594326054423, −15.777356751970020186497250334055, −14.51515742982798714676552306271, −14.04336651019214150624188211036, −13.019179402138733444886309004621, −12.28408234387777985569967431899, −11.17319502534160625254065565136, −9.50934692135668510855257200033, −8.37778245809955504506688678807, −7.218243877192699348114302576054, −6.35959036172102173828773838593, −4.78869095350382788978369275414, −4.02269955237858526095341021489, −2.1657511557552747608955488014, −1.49198813267859622444579542563, 2.02342496271580856112835741136, 2.84047785117219985170580562766, 4.0364753412728539581568168863, 5.23909551234790251930325147002, 6.256560017591392382053949881503, 7.690466068402902946495055265729, 8.93918392605450745918635656848, 10.35992293606220144505040617199, 10.96089593212277738248500868888, 12.19765984725759132084424004232, 13.62160775730238312765144682567, 14.35105087858839561231969769258, 14.93477408996356359271221235643, 15.84625421376726485255958531869, 17.13865843259677613562375076294, 18.521708269556160974901734375945, 19.58506595118768652058587372969, 20.580348492455981866944206905891, 21.438201107528310156464451590, 22.09242318775518108418248498547, 22.73205011471550728077381497310, 24.43724157569836424428707781390, 24.90889234679199433906195896520, 25.82620106112979275029331798969, 27.00193159674936807078811584933

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