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Properties

Label 1-191-191.52-r0-0-0
Degree $1$
Conductor $191$
Sign $0.209 + 0.977i$
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

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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

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.209 + 0.977i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ 0.209 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8929113659 + 0.7219110450i\)
\(L(\frac12)\) \(\approx\) \(0.8929113659 + 0.7219110450i\)
\(L(1)\) \(\approx\) \(0.9281299419 + 0.5030652151i\)
\(L(1)\) \(\approx\) \(0.9281299419 + 0.5030652151i\)

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

−26.76326514803028552694595748696, −26.29872227883871705019491608185, −25.21079883030972692655246843456, −23.84270639070581518807126049071, −23.20842713255312266319360895233, −21.84780058209901777173366796619, −20.71792518089597231363971591479, −20.34223487093969228569400669096, −18.92001467255373022867978580038, −18.55605443862689855256439117512, −17.84202720643579266280413465458, −16.21848017938332528784092683198, −14.9019690696497561545455615984, −13.95897344494597215075659326191, −13.120920943560122058598536059329, −11.72452525914383052442164854769, −11.2412020811132350132839451841, −9.927000576848564780051476574334, −8.61324314076143678105083156708, −7.89126349500940598908358496454, −7.02230747917000094313079727684, −4.85939108897874235769600168042, −3.4368376914094445536646646034, −2.629518212496179441062450901, −1.22243271440988846605226920769, 1.434504214961270737835504291921, 3.54713509123782100212660093900, 4.78511365996479253243600796187, 5.4834544603396510935286147429, 7.61331390663205956295549439537, 8.067231197373386225643134951908, 8.86385712629729371176908174002, 10.112821571463435775735206918187, 11.10112567228661190804219834309, 12.85882327841492874354900437460, 13.843024329091482009590512390639, 14.96483292232290651900352753393, 15.577285588680929218539970650673, 16.36702535964209647980801401290, 17.4980463886497547486072880, 18.589356601075143330224169693397, 19.59221054495241111497835585621, 20.61791586325968963069378993572, 21.26975051052199802155797898412, 22.9086786973985479610818707795, 23.673569904483624808494832285257, 24.66173042801026603092594624977, 25.32955747727084633128014882718, 26.47622509672966035773854473928, 27.09782699233296191515403149587

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