↑

Properties

Label 1-191-191.25-r0-0-0
Degree $1$
Conductor $191$
Sign $-0.432 + 0.901i$
Analytic cond. $0.887000$
Root an. cond. $0.887000$
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  + (0.546 + 0.837i)2-s + (0.945 + 0.324i)3-s + (−0.401 + 0.915i)4-s + (−0.401 + 0.915i)5-s + (0.245 + 0.969i)6-s + 7-s + (−0.986 + 0.164i)8-s + (0.789 + 0.614i)9-s + (−0.986 + 0.164i)10-s + (−0.0825 − 0.996i)11-s + (−0.677 + 0.735i)12-s + (0.945 − 0.324i)13-s + (0.546 + 0.837i)14-s + (−0.677 + 0.735i)15-s + (−0.677 − 0.735i)16-s + (−0.879 − 0.475i)17-s + ⋯
L(s)  = 1  + (0.546 + 0.837i)2-s + (0.945 + 0.324i)3-s + (−0.401 + 0.915i)4-s + (−0.401 + 0.915i)5-s + (0.245 + 0.969i)6-s + 7-s + (−0.986 + 0.164i)8-s + (0.789 + 0.614i)9-s + (−0.986 + 0.164i)10-s + (−0.0825 − 0.996i)11-s + (−0.677 + 0.735i)12-s + (0.945 − 0.324i)13-s + (0.546 + 0.837i)14-s + (−0.677 + 0.735i)15-s + (−0.677 − 0.735i)16-s + (−0.879 − 0.475i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.432 + 0.901i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ -0.432 + 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.037250962 + 1.647746552i\)
\(L(\frac12)\) \(\approx\) \(1.037250962 + 1.647746552i\)
\(L(1)\) \(\approx\) \(1.294460099 + 1.102014519i\)
\(L(1)\) \(\approx\) \(1.294460099 + 1.102014519i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.546 + 0.837i)T \)
3 \( 1 + (0.945 + 0.324i)T \)
5 \( 1 + (-0.401 + 0.915i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.0825 - 0.996i)T \)
13 \( 1 + (0.945 - 0.324i)T \)
17 \( 1 + (-0.879 - 0.475i)T \)
19 \( 1 + (-0.986 - 0.164i)T \)
23 \( 1 + (-0.986 - 0.164i)T \)
29 \( 1 + (-0.677 + 0.735i)T \)
31 \( 1 + (0.789 + 0.614i)T \)
37 \( 1 + (0.789 - 0.614i)T \)
41 \( 1 + (0.546 - 0.837i)T \)
43 \( 1 + (0.245 + 0.969i)T \)
47 \( 1 + (-0.0825 - 0.996i)T \)
53 \( 1 + (-0.0825 - 0.996i)T \)
59 \( 1 + (0.789 + 0.614i)T \)
61 \( 1 + (-0.879 + 0.475i)T \)
67 \( 1 + (-0.879 + 0.475i)T \)
71 \( 1 + (0.546 - 0.837i)T \)
73 \( 1 + (-0.0825 + 0.996i)T \)
79 \( 1 + (-0.677 - 0.735i)T \)
83 \( 1 + (-0.986 + 0.164i)T \)
89 \( 1 + (0.245 - 0.969i)T \)
97 \( 1 + (0.789 - 0.614i)T \)
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

−26.99529509590882545447989312674, −25.75145651144598742552386880888, −24.56260805624771405403637370211, −23.93698292744912768758541391978, −23.19111287309248182280400932751, −21.63190966259273936378863476562, −20.72501502118156348323898256183, −20.366474023406079844376226326410, −19.406799047134332549948818832049, −18.39909156455102635935103879848, −17.39180092580418313075292710429, −15.55795516679964414055970234402, −14.953300603118808769804033309, −13.79063044971118365511118248169, −13.00648887412499197618267563741, −12.12880899680241843502721132843, −11.09587596256800838660639552578, −9.70340616079306765920548214260, −8.692323337165270680652036279139, −7.87410144480624702663536501044, −6.154335177361281883033170581, −4.42460091149220602493441884830, −4.11740209466059314337441430154, −2.231476740586793521071866227279, −1.4369684443247684608252186261, 2.44157521691649809043904535826, 3.59644055696229067787123652732, 4.474256391468748653315692324779, 5.97674602595535463596878349970, 7.21128084537292604216478442100, 8.21825076522225404489407332494, 8.78844346005479006895047683097, 10.60395647032659023089179324271, 11.4837907163840151771647566281, 13.130071670781830591412662151325, 14.01319336505083657222740408925, 14.6512712213310654685740770823, 15.54039679300529747311912622711, 16.26853205821136201190476695026, 17.86096265914953441813273554518, 18.529271506907674445645038902375, 19.80880128759832697981031749422, 21.03240587727874291957318937030, 21.65047718374423115236731145987, 22.652617440958779867965856953200, 23.811371193286918836781133149286, 24.49720760795539277400408488283, 25.57210573626424462736835403592, 26.34098322037706224356639430924, 27.03462067629794930124779011215

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