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Properties

Label 1-191-191.70-r1-0-0
Degree $1$
Conductor $191$
Sign $-0.317 + 0.948i$
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  + (−0.677 + 0.735i)2-s + (0.245 − 0.969i)3-s + (−0.0825 − 0.996i)4-s + (−0.0825 − 0.996i)5-s + (0.546 + 0.837i)6-s − 7-s + (0.789 + 0.614i)8-s + (−0.879 − 0.475i)9-s + (0.789 + 0.614i)10-s + (−0.945 − 0.324i)11-s + (−0.986 − 0.164i)12-s + (0.245 + 0.969i)13-s + (0.677 − 0.735i)14-s + (−0.986 − 0.164i)15-s + (−0.986 + 0.164i)16-s + (−0.401 − 0.915i)17-s + ⋯
L(s)  = 1  + (−0.677 + 0.735i)2-s + (0.245 − 0.969i)3-s + (−0.0825 − 0.996i)4-s + (−0.0825 − 0.996i)5-s + (0.546 + 0.837i)6-s − 7-s + (0.789 + 0.614i)8-s + (−0.879 − 0.475i)9-s + (0.789 + 0.614i)10-s + (−0.945 − 0.324i)11-s + (−0.986 − 0.164i)12-s + (0.245 + 0.969i)13-s + (0.677 − 0.735i)14-s + (−0.986 − 0.164i)15-s + (−0.986 + 0.164i)16-s + (−0.401 − 0.915i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.317 + 0.948i$
Analytic conductor: \(20.5258\)
Root analytic conductor: \(20.5258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ -0.317 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06837306867 + 0.09496765952i\)
\(L(\frac12)\) \(\approx\) \(0.06837306867 + 0.09496765952i\)
\(L(1)\) \(\approx\) \(0.5504825316 - 0.1135242554i\)
\(L(1)\) \(\approx\) \(0.5504825316 - 0.1135242554i\)

Euler product

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

−26.595177206079499814647008478574, −25.81524558031922603364915574172, −25.48340475849451017854252432366, −23.21525962051726669975237156876, −22.49494866927991304734251626868, −21.66630111991621256977064889629, −20.84277693343662853217978417412, −19.66540147789511929854823244238, −19.18448152773620672739740630260, −17.91123530508203562046756835413, −17.05331952762412050516632384204, −15.64003882442602286271884463870, −15.30695470302875523233139138951, −13.60069005715649393896140638134, −12.70800073551611198532652661935, −11.157129702832980424541314816132, −10.462644450998810779345209297323, −9.86639061689481267826022921933, −8.65108195008261749073554804444, −7.567114922085762174568464828837, −6.143525100080467206384466215556, −4.36013970262572827062183884889, −3.14697819505538280636775716084, −2.57888181454791413956437061318, −0.05644071349071664743309640791, 1.07165209189452243262238040585, 2.59825426221860802390802813926, 4.66431749780929996065660720152, 6.02848292094459268703508818497, 6.81870334610863337588899603392, 8.04623685284106865120189525484, 8.80415552066570297976181718053, 9.72459083805883564342374060317, 11.25258106906615544219102919831, 12.63424540091558996698421060182, 13.37198901138446721663951623388, 14.367964179361216185637383740838, 15.88271843504217107180504611510, 16.39873506593225829159944990936, 17.45730620447277937582195418951, 18.56240451860793081953439317716, 19.210013532574868220441286222336, 20.05343596596462397009054725395, 21.14491776676729466077668687513, 23.12622647400681593646564638397, 23.40388450893946672070516818179, 24.59191543926976030526196714934, 25.081364646580500821462044998904, 26.06823191056700263413483065127, 26.84712134968651788716808929465

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