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Properties

Label 1-293-293.278-r0-0-0
Degree $1$
Conductor $293$
Sign $0.834 - 0.551i$
Analytic cond. $1.36068$
Root an. cond. $1.36068$
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  + (−0.683 − 0.729i)2-s + (0.908 − 0.417i)3-s + (−0.0645 + 0.997i)4-s + (0.996 + 0.0859i)5-s + (−0.925 − 0.377i)6-s + (0.192 + 0.981i)7-s + (0.772 − 0.635i)8-s + (0.651 − 0.758i)9-s + (−0.618 − 0.785i)10-s + (−0.976 + 0.213i)11-s + (0.357 + 0.933i)12-s + (0.714 + 0.699i)13-s + (0.584 − 0.811i)14-s + (0.941 − 0.337i)15-s + (−0.991 − 0.128i)16-s + (−0.683 − 0.729i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.729i)2-s + (0.908 − 0.417i)3-s + (−0.0645 + 0.997i)4-s + (0.996 + 0.0859i)5-s + (−0.925 − 0.377i)6-s + (0.192 + 0.981i)7-s + (0.772 − 0.635i)8-s + (0.651 − 0.758i)9-s + (−0.618 − 0.785i)10-s + (−0.976 + 0.213i)11-s + (0.357 + 0.933i)12-s + (0.714 + 0.699i)13-s + (0.584 − 0.811i)14-s + (0.941 − 0.337i)15-s + (−0.991 − 0.128i)16-s + (−0.683 − 0.729i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $0.834 - 0.551i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ 0.834 - 0.551i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.369367785 - 0.4118140988i\)
\(L(\frac12)\) \(\approx\) \(1.369367785 - 0.4118140988i\)
\(L(1)\) \(\approx\) \(1.147939104 - 0.3175228547i\)
\(L(1)\) \(\approx\) \(1.147939104 - 0.3175228547i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (-0.683 - 0.729i)T \)
3 \( 1 + (0.908 - 0.417i)T \)
5 \( 1 + (0.996 + 0.0859i)T \)
7 \( 1 + (0.192 + 0.981i)T \)
11 \( 1 + (-0.976 + 0.213i)T \)
13 \( 1 + (0.714 + 0.699i)T \)
17 \( 1 + (-0.683 - 0.729i)T \)
19 \( 1 + (0.941 - 0.337i)T \)
23 \( 1 + (-0.925 + 0.377i)T \)
29 \( 1 + (-0.234 + 0.972i)T \)
31 \( 1 + (0.996 - 0.0859i)T \)
37 \( 1 + (0.966 + 0.255i)T \)
41 \( 1 + (-0.548 + 0.835i)T \)
43 \( 1 + (-0.474 - 0.880i)T \)
47 \( 1 + (0.823 - 0.566i)T \)
53 \( 1 + (-0.976 + 0.213i)T \)
59 \( 1 + (-0.744 - 0.668i)T \)
61 \( 1 + (0.436 + 0.899i)T \)
67 \( 1 + (0.107 - 0.994i)T \)
71 \( 1 + (-0.798 - 0.601i)T \)
73 \( 1 + (-0.0645 - 0.997i)T \)
79 \( 1 + (0.436 + 0.899i)T \)
83 \( 1 + (0.276 - 0.961i)T \)
89 \( 1 + (0.436 - 0.899i)T \)
97 \( 1 + (-0.317 - 0.948i)T \)
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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

−25.73649600847024506455015642927, −24.80751922633299978467328054906, −24.13459868931556026348573487164, −23.01573190921945178509359826357, −21.8544871366292782059565808703, −20.533738446076568122486837715280, −20.39979141023543586037381507448, −19.07784121744420705693071009385, −18.11123483449034371545378934434, −17.38637942890831005220987309581, −16.2827443621734726551420559029, −15.6006559687377374942164935787, −14.50260897847114156605562911483, −13.659186632895251483605994482460, −13.204589556900519297716049392048, −10.83723067833732272209133926589, −10.263611345149657594831619762130, −9.5239810827375931178897112844, −8.27915915207957842399454853530, −7.802396817588902490609833662492, −6.415990782488944733001128401396, −5.35656171322716127103361009380, −4.168922040396671806761481667347, −2.56302209143622603709255915589, −1.29411303713614248694086835209, 1.53376142124470000204839249858, 2.36000474009144777560104156741, 3.15047279507926290822597742104, 4.825006376250191488655585458942, 6.356991247487279816227524787967, 7.52495756175847118687566138274, 8.59664252743442426779724416887, 9.2737348528192624495738788050, 10.03941774798200184387182440266, 11.35427354137262790921135767089, 12.343372852768245804958719414114, 13.40452809856164371306923774265, 13.85739302628521843299187661736, 15.3372365702151051376785466249, 16.22646926122619604089502876800, 17.77944589689403254214490373888, 18.28497072445431638606639320477, 18.739164346213674571596260431176, 20.09013224598988077791479838822, 20.70328073448151406601443585632, 21.5224004093629754733810548608, 22.173865801772340575302895158391, 23.81561236141757538913027512648, 24.888464105841301632429550078471, 25.51005619102719500635575816146

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