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Properties

Label 1-101-101.70-r0-0-0
Degree $1$
Conductor $101$
Sign $-0.769 + 0.638i$
Analytic cond. $0.469042$
Root an. cond. $0.469042$
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  + (−0.535 + 0.844i)2-s + (−0.968 + 0.248i)3-s + (−0.425 − 0.904i)4-s + (0.535 + 0.844i)5-s + (0.309 − 0.951i)6-s + (0.929 + 0.368i)7-s + (0.992 + 0.125i)8-s + (0.876 − 0.481i)9-s − 10-s + (−0.876 + 0.481i)11-s + (0.637 + 0.770i)12-s + (−0.929 + 0.368i)13-s + (−0.809 + 0.587i)14-s + (−0.728 − 0.684i)15-s + (−0.637 + 0.770i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (−0.535 + 0.844i)2-s + (−0.968 + 0.248i)3-s + (−0.425 − 0.904i)4-s + (0.535 + 0.844i)5-s + (0.309 − 0.951i)6-s + (0.929 + 0.368i)7-s + (0.992 + 0.125i)8-s + (0.876 − 0.481i)9-s − 10-s + (−0.876 + 0.481i)11-s + (0.637 + 0.770i)12-s + (−0.929 + 0.368i)13-s + (−0.809 + 0.587i)14-s + (−0.728 − 0.684i)15-s + (−0.637 + 0.770i)16-s + (0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $-0.769 + 0.638i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ -0.769 + 0.638i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1921762600 + 0.5326159189i\)
\(L(\frac12)\) \(\approx\) \(0.1921762600 + 0.5326159189i\)
\(L(1)\) \(\approx\) \(0.4861107243 + 0.4200512359i\)
\(L(1)\) \(\approx\) \(0.4861107243 + 0.4200512359i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (-0.535 + 0.844i)T \)
3 \( 1 + (-0.968 + 0.248i)T \)
5 \( 1 + (0.535 + 0.844i)T \)
7 \( 1 + (0.929 + 0.368i)T \)
11 \( 1 + (-0.876 + 0.481i)T \)
13 \( 1 + (-0.929 + 0.368i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.637 - 0.770i)T \)
23 \( 1 + (0.0627 + 0.998i)T \)
29 \( 1 + (0.929 - 0.368i)T \)
31 \( 1 + (-0.929 - 0.368i)T \)
37 \( 1 + (0.968 + 0.248i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + (-0.187 + 0.982i)T \)
47 \( 1 + (-0.187 - 0.982i)T \)
53 \( 1 + (0.425 - 0.904i)T \)
59 \( 1 + (0.637 - 0.770i)T \)
61 \( 1 + (0.425 + 0.904i)T \)
67 \( 1 + (-0.968 - 0.248i)T \)
71 \( 1 + (0.968 - 0.248i)T \)
73 \( 1 + (-0.0627 - 0.998i)T \)
79 \( 1 + (0.0627 - 0.998i)T \)
83 \( 1 + (-0.0627 + 0.998i)T \)
89 \( 1 + (0.637 + 0.770i)T \)
97 \( 1 + (-0.425 - 0.904i)T \)
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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

−29.23923017300321711852587619118, −28.7356114790547018142841978394, −27.4442273177673667274803586341, −27.05836389178573219921504594849, −25.29799385697970506872639938495, −24.27312470237140823846533722138, −23.20845546133651888699894277663, −21.884649945584086650984699543290, −21.07900408017184461987274290133, −20.22123593202689524124959950616, −18.68752492704263367183619359249, −17.86789407093350559078722537318, −16.99829198238748327534525076785, −16.24051742075911722988579506129, −14.03659643290500269207153253231, −12.82091429391675981233926039573, −12.09331714961388098889161564272, −10.81593875402129366824732771151, −10.0462451600334944285335660363, −8.48352955082973503304460609236, −7.40347889598456503569203798780, −5.39737968793019845515766844366, −4.52835519337542155067160068568, −2.23737505494861895849117315970, −0.79553175515607210391539538115, 1.9413406233168769801388590740, 4.66685820329864362848018803198, 5.6112739151385418721982284384, 6.75806692149256366467758274163, 7.85918982790299953090294939686, 9.5832723989618470419231214664, 10.45929166671221080612010064780, 11.46062869344126305407584417592, 13.17204778409654582014874724743, 14.78993372250469339161860492181, 15.213196276909863664198729007741, 16.724139815127618053147077869674, 17.69450187993225765037505069540, 18.121741846657073306543923685035, 19.35166939813557601726623425395, 21.31683810839996473290039870275, 21.99313199934335448088996768642, 23.35430917465265538615237471229, 23.92453791917426427481448045257, 25.22365458340277134296776026414, 26.285201099548747068675634857519, 27.14398426774792904050381757672, 28.12531244193843866893435313405, 28.94885346335364241949827533729, 30.112126142776407245826342693498

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