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Properties

Label 1-131-131.16-r0-0-0
Degree $1$
Conductor $131$
Sign $0.194 + 0.980i$
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  + (0.981 + 0.192i)2-s + (0.215 + 0.976i)3-s + (0.926 + 0.377i)4-s + (−0.861 + 0.506i)5-s + (0.0241 + 0.999i)6-s + (0.958 − 0.285i)7-s + (0.836 + 0.548i)8-s + (−0.906 + 0.421i)9-s + (−0.943 + 0.331i)10-s + (−0.168 − 0.985i)11-s + (−0.168 + 0.985i)12-s + (−0.943 − 0.331i)13-s + (0.995 − 0.0965i)14-s + (−0.681 − 0.732i)15-s + (0.715 + 0.698i)16-s + (−0.443 + 0.896i)17-s + ⋯
L(s)  = 1  + (0.981 + 0.192i)2-s + (0.215 + 0.976i)3-s + (0.926 + 0.377i)4-s + (−0.861 + 0.506i)5-s + (0.0241 + 0.999i)6-s + (0.958 − 0.285i)7-s + (0.836 + 0.548i)8-s + (−0.906 + 0.421i)9-s + (−0.943 + 0.331i)10-s + (−0.168 − 0.985i)11-s + (−0.168 + 0.985i)12-s + (−0.943 − 0.331i)13-s + (0.995 − 0.0965i)14-s + (−0.681 − 0.732i)15-s + (0.715 + 0.698i)16-s + (−0.443 + 0.896i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $0.194 + 0.980i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ 0.194 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.427737055 + 1.172311202i\)
\(L(\frac12)\) \(\approx\) \(1.427737055 + 1.172311202i\)
\(L(1)\) \(\approx\) \(1.541327977 + 0.7771967224i\)
\(L(1)\) \(\approx\) \(1.541327977 + 0.7771967224i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (0.981 + 0.192i)T \)
3 \( 1 + (0.215 + 0.976i)T \)
5 \( 1 + (-0.861 + 0.506i)T \)
7 \( 1 + (0.958 - 0.285i)T \)
11 \( 1 + (-0.168 - 0.985i)T \)
13 \( 1 + (-0.943 - 0.331i)T \)
17 \( 1 + (-0.443 + 0.896i)T \)
19 \( 1 + (0.885 + 0.464i)T \)
23 \( 1 + (-0.262 - 0.964i)T \)
29 \( 1 + (-0.906 - 0.421i)T \)
31 \( 1 + (0.779 - 0.626i)T \)
37 \( 1 + (-0.0724 + 0.997i)T \)
41 \( 1 + (0.715 - 0.698i)T \)
43 \( 1 + (0.399 - 0.916i)T \)
47 \( 1 + (0.120 + 0.992i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.989 - 0.144i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.399 + 0.916i)T \)
71 \( 1 + (-0.354 - 0.935i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.748 - 0.663i)T \)
83 \( 1 + (-0.527 + 0.849i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.607 + 0.794i)T \)
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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

−28.64830037953726049760681763043, −27.81446554240286227356067296721, −26.321211181182738972867921784284, −24.68471510999946737290540880393, −24.60458444410686096562110788422, −23.49953839998020740053800794696, −22.7707573482081543026183941741, −21.38262829682481871341009082552, −20.13583868807976034382676969670, −19.84568777582931441037146100628, −18.44664216598007183743640264532, −17.30263012550997429922904200932, −15.79783196652579580491582598855, −14.85006780551582824219289909163, −13.92678976365637393518791522221, −12.723871655872693387262738608092, −11.92186857612428475552958681494, −11.31192608710935942429394254316, −9.28589679072677077290011637226, −7.6304693775219150252471769280, −7.207484124913736400422130389825, −5.36688104126253716299325944249, −4.47095358628278599907537941344, −2.7711938211190377674332145795, −1.527324227638635191763199535408, 2.609977725950688205345889924784, 3.785167951462365184023836170489, 4.655101438281449025495412393242, 5.886990216647363982621424341277, 7.58765149182022818549959516982, 8.34907318874653860117177445637, 10.39420034199352720611890289665, 11.14332392126406556126237898422, 12.05770017038048560272631099024, 13.750708208181711951644939085472, 14.63594205863176207845086031963, 15.26755102994563160688567571533, 16.29712051480356600175848169625, 17.2791111222065394261906786207, 19.10304738783183097590789366564, 20.2015606213784046087407725280, 20.94310427783434014587499564058, 22.10020650553907427027289728754, 22.59828220366031096385706195634, 23.968853681345633044855204550319, 24.53687065170446573778979322388, 26.193308161988686738538097187813, 26.73552785792584858599659177559, 27.68439723080575209611600250033, 29.04461249606561555588956027071

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