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Properties

Label 1-131-131.3-r0-0-0
Degree $1$
Conductor $131$
Sign $0.695 - 0.718i$
Analytic cond. $0.608361$
Root an. cond. $0.608361$
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  + (−0.943 − 0.331i)2-s + (0.715 − 0.698i)3-s + (0.779 + 0.626i)4-s + (−0.989 + 0.144i)5-s + (−0.906 + 0.421i)6-s + (0.485 + 0.873i)7-s + (−0.527 − 0.849i)8-s + (0.0241 − 0.999i)9-s + (0.981 + 0.192i)10-s + (0.995 + 0.0965i)11-s + (0.995 − 0.0965i)12-s + (0.981 − 0.192i)13-s + (−0.168 − 0.985i)14-s + (−0.607 + 0.794i)15-s + (0.215 + 0.976i)16-s + (0.399 − 0.916i)17-s + ⋯
L(s)  = 1  + (−0.943 − 0.331i)2-s + (0.715 − 0.698i)3-s + (0.779 + 0.626i)4-s + (−0.989 + 0.144i)5-s + (−0.906 + 0.421i)6-s + (0.485 + 0.873i)7-s + (−0.527 − 0.849i)8-s + (0.0241 − 0.999i)9-s + (0.981 + 0.192i)10-s + (0.995 + 0.0965i)11-s + (0.995 − 0.0965i)12-s + (0.981 − 0.192i)13-s + (−0.168 − 0.985i)14-s + (−0.607 + 0.794i)15-s + (0.215 + 0.976i)16-s + (0.399 − 0.916i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $0.695 - 0.718i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ 0.695 - 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7899833178 - 0.3347737017i\)
\(L(\frac12)\) \(\approx\) \(0.7899833178 - 0.3347737017i\)
\(L(1)\) \(\approx\) \(0.8195419203 - 0.2371438199i\)
\(L(1)\) \(\approx\) \(0.8195419203 - 0.2371438199i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (-0.943 - 0.331i)T \)
3 \( 1 + (0.715 - 0.698i)T \)
5 \( 1 + (-0.989 + 0.144i)T \)
7 \( 1 + (0.485 + 0.873i)T \)
11 \( 1 + (0.995 + 0.0965i)T \)
13 \( 1 + (0.981 - 0.192i)T \)
17 \( 1 + (0.399 - 0.916i)T \)
19 \( 1 + (-0.748 + 0.663i)T \)
23 \( 1 + (-0.0724 - 0.997i)T \)
29 \( 1 + (0.0241 + 0.999i)T \)
31 \( 1 + (0.926 + 0.377i)T \)
37 \( 1 + (-0.262 - 0.964i)T \)
41 \( 1 + (0.215 - 0.976i)T \)
43 \( 1 + (-0.443 - 0.896i)T \)
47 \( 1 + (0.568 + 0.822i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.861 + 0.506i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.443 + 0.896i)T \)
71 \( 1 + (-0.970 + 0.239i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.885 + 0.464i)T \)
83 \( 1 + (0.836 + 0.548i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.681 + 0.732i)T \)
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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

−28.027592377998196626034201837858, −27.8357534992996165355250164344, −26.75785526475937403530804210649, −26.16009996474611672531310422799, −25.07275228791954618251263988474, −23.961455240470652141310576191882, −23.12783042984483901519994266818, −21.39865125108496972447177866464, −20.43581819097457772871081918846, −19.61821613354424530373350239460, −19.01678701724266029962604504522, −17.31580160557073408068331082397, −16.57787535161914696927685607196, −15.486091176063583006991707174458, −14.78996626666671903881701790949, −13.59920409039776866031225923664, −11.56230132641391756857094013343, −10.842512260045060965680829688447, −9.6337805195087358217509979597, −8.434089787687392126629780637434, −7.88431077608394471539034363514, −6.490836855935215972943170260144, −4.53432807544816681473370454920, −3.48692130250269904330807330863, −1.44043437305641903428100275278, 1.27265085865692262557948663173, 2.70253057052294961039177189891, 3.88831509217275532723937968610, 6.33673839114117561807720505113, 7.44046383453780770810748804690, 8.50440210880984828690364503259, 8.9986144413485270215948830861, 10.75584611548996757659180022022, 11.977400653393735131153211443843, 12.407988833867507866508069464138, 14.25284789052797521864493226673, 15.24897366048496485868598805911, 16.279705517610634597643550477674, 17.75962207681891509155005519902, 18.66704526539836461568158013026, 19.181932756072961490289996590223, 20.30252955924647954919504789123, 20.99392101291200748354803193946, 22.54537544223232261055167622741, 23.87262916649708166579141896570, 24.95064955816719402532513569798, 25.4507317921004541791114178840, 26.742493627532155349628743759041, 27.535317661665106187882158436765, 28.32626351355785231487611102080

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