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Properties

Label 1-191-191.139-r1-0-0
Degree $1$
Conductor $191$
Sign $-0.421 + 0.906i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.401 + 0.915i)2-s + (0.789 + 0.614i)3-s + (−0.677 − 0.735i)4-s + (−0.677 − 0.735i)5-s + (−0.879 + 0.475i)6-s − 7-s + (0.945 − 0.324i)8-s + (0.245 + 0.969i)9-s + (0.945 − 0.324i)10-s + (0.986 − 0.164i)11-s + (−0.0825 − 0.996i)12-s + (0.789 − 0.614i)13-s + (0.401 − 0.915i)14-s + (−0.0825 − 0.996i)15-s + (−0.0825 + 0.996i)16-s + (0.546 + 0.837i)17-s + ⋯
L(s)  = 1  + (−0.401 + 0.915i)2-s + (0.789 + 0.614i)3-s + (−0.677 − 0.735i)4-s + (−0.677 − 0.735i)5-s + (−0.879 + 0.475i)6-s − 7-s + (0.945 − 0.324i)8-s + (0.245 + 0.969i)9-s + (0.945 − 0.324i)10-s + (0.986 − 0.164i)11-s + (−0.0825 − 0.996i)12-s + (0.789 − 0.614i)13-s + (0.401 − 0.915i)14-s + (−0.0825 − 0.996i)15-s + (−0.0825 + 0.996i)16-s + (0.546 + 0.837i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.421 + 0.906i$
Analytic conductor: \(20.5258\)
Root analytic conductor: \(20.5258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ -0.421 + 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7643561088 + 1.198590077i\)
\(L(\frac12)\) \(\approx\) \(0.7643561088 + 1.198590077i\)
\(L(1)\) \(\approx\) \(0.8149290544 + 0.5149121800i\)
\(L(1)\) \(\approx\) \(0.8149290544 + 0.5149121800i\)

Euler product

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

−26.60751010977811788026397536760, −25.7416733746811970820682537278, −25.05794125682283817591831272251, −23.260880894366725079053898042147, −22.88563696932567720672465169289, −21.59365232605430458679487535533, −20.5314162465565799538091752720, −19.55834346739638200761789276461, −19.02056413488112573371888745449, −18.44743673521820947082800613922, −17.081416717932800573797058491, −15.847836305797543027985128798022, −14.49582830771506924368941709925, −13.68540903013664694675338771683, −12.52494811076324050430397428902, −11.81149433410286190950105824810, −10.59436420243639017384017368554, −9.37726643426720627824319277141, −8.636913779584913325705014675926, −7.32365102307341082901911854767, −6.53424071962015360676385607871, −3.99420717431458061755464404299, −3.36450370121796692167498117571, −2.19677871421266718169118590385, −0.64005487386173877617399164786, 1.122401390669625379038538071665, 3.44641584271793576644628006199, 4.29668526745059014622402867281, 5.675526538820626077704156819743, 6.960575492090870813198686299317, 8.26789890773705700662467974440, 8.844815557690013232519728750489, 9.76553190313494344265581788872, 10.92997032899329249657654477997, 12.76637842864361584786952017003, 13.51790481163335533951850346191, 14.94047929006398092365781919098, 15.3961336886413355651307323290, 16.54906287442275086648508858120, 16.89672065761998845375344717739, 18.709336718687296593293839439443, 19.527780473984879285873067149004, 20.05487566482673665785512436647, 21.4624957656296882843593609647, 22.61162915928783101017660094827, 23.46093423403772420933352793702, 24.57510436190700209567914749160, 25.49299006408664140338402107828, 25.93756362986070067724474929615, 27.234724919642493084628103037942

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