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Properties

Label 1-931-931.141-r1-0-0
Degree $1$
Conductor $931$
Sign $0.0303 + 0.999i$
Analytic cond. $100.049$
Root an. cond. $100.049$
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  + (0.733 − 0.680i)2-s + (−0.365 + 0.930i)3-s + (0.0747 − 0.997i)4-s + (0.365 − 0.930i)5-s + (0.365 + 0.930i)6-s + (−0.623 − 0.781i)8-s + (−0.733 − 0.680i)9-s + (−0.365 − 0.930i)10-s + (−0.222 − 0.974i)11-s + (0.900 + 0.433i)12-s + (−0.955 − 0.294i)13-s + (0.733 + 0.680i)15-s + (−0.988 − 0.149i)16-s + (0.0747 + 0.997i)17-s − 18-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)2-s + (−0.365 + 0.930i)3-s + (0.0747 − 0.997i)4-s + (0.365 − 0.930i)5-s + (0.365 + 0.930i)6-s + (−0.623 − 0.781i)8-s + (−0.733 − 0.680i)9-s + (−0.365 − 0.930i)10-s + (−0.222 − 0.974i)11-s + (0.900 + 0.433i)12-s + (−0.955 − 0.294i)13-s + (0.733 + 0.680i)15-s + (−0.988 − 0.149i)16-s + (0.0747 + 0.997i)17-s − 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0303 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0303 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.0303 + 0.999i$
Analytic conductor: \(100.049\)
Root analytic conductor: \(100.049\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 931,\ (1:\ ),\ 0.0303 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05758817885 + 0.05586715858i\)
\(L(\frac12)\) \(\approx\) \(0.05758817885 + 0.05586715858i\)
\(L(1)\) \(\approx\) \(0.9864564687 - 0.4649192251i\)
\(L(1)\) \(\approx\) \(0.9864564687 - 0.4649192251i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.733 - 0.680i)T \)
3 \( 1 + (-0.365 + 0.930i)T \)
5 \( 1 + (0.365 - 0.930i)T \)
11 \( 1 + (-0.222 - 0.974i)T \)
13 \( 1 + (-0.955 - 0.294i)T \)
17 \( 1 + (0.0747 + 0.997i)T \)
23 \( 1 + (0.0747 - 0.997i)T \)
29 \( 1 + (-0.826 + 0.563i)T \)
31 \( 1 - T \)
37 \( 1 + (0.900 + 0.433i)T \)
41 \( 1 + (-0.365 + 0.930i)T \)
43 \( 1 + (-0.988 - 0.149i)T \)
47 \( 1 + (0.955 + 0.294i)T \)
53 \( 1 + (-0.0747 + 0.997i)T \)
59 \( 1 + (0.988 + 0.149i)T \)
61 \( 1 + (0.826 - 0.563i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.826 - 0.563i)T \)
73 \( 1 + (0.955 - 0.294i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.733 + 0.680i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−21.95982004365261191541901183846, −20.79938663421748002152284901065, −19.90295478286584017729115772117, −18.86029408432073110461460618294, −18.09749356671613119129324220891, −17.50906958674480152890634664827, −16.85286861460946116767972806997, −15.80578329462302738420774008405, −14.79776865362352839318679852607, −14.36685748502549514856736266966, −13.410583414692628795697319564197, −12.88018933203166917224408689934, −11.79541598435292617964206782712, −11.42583322343228891017768592350, −10.10204851441198970680193174085, −9.124911718279301654842125693294, −7.65666014015794659406404428311, −7.29807296037731828297192856715, −6.71202464788318940776677428196, −5.61932108481689668106369821389, −5.07291066462892141735823883152, −3.74025689377398831003994235009, −2.53762974672628613703137993908, −2.01153393980565710563011463283, −0.01436572913266088293751385095, 0.99203936073459196595656005314, 2.30767360900737167467844910572, 3.379514571436758150870503320836, 4.24240608393416723851354156172, 5.096152350408945203111651076171, 5.63490321773745537923576488557, 6.47889855921631088448899016024, 8.18851000215587067132525637677, 9.08952161319126422125646710540, 9.82393987014800602349455059793, 10.60756061579658568944760304559, 11.2896835588643101387131542798, 12.313895348880947146631871920213, 12.82511835468398473704378634813, 13.79487702425819041693133414435, 14.72324072808606413731390000410, 15.2745686374315793768998857510, 16.50764496648486253606343890320, 16.69837842403254816490477846090, 17.88322460747438555066385433435, 18.87985823780204651485630298214, 19.95646543220125954736534030938, 20.35371655107003768493536481603, 21.20569972481592197627164383183, 21.93092637350142854735529912925

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