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Properties

Label 1-191-191.11-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: $\chi_{191} (11, \cdot )$
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

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

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