Properties

Degree 1
Conductor $ 2^{6} \cdot 5 $
Sign $-0.998 + 0.0626i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.382 + 0.923i)3-s + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.923 − 0.382i)11-s + (−0.923 + 0.382i)13-s − 17-s + (−0.382 − 0.923i)19-s + (−0.923 − 0.382i)21-s + (0.707 + 0.707i)23-s + (−0.923 − 0.382i)27-s + (−0.923 + 0.382i)29-s + 31-s i·33-s + (0.923 + 0.382i)37-s + (−0.707 − 0.707i)39-s + ⋯
L(s,χ)  = 1  + (0.382 + 0.923i)3-s + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.923 − 0.382i)11-s + (−0.923 + 0.382i)13-s − 17-s + (−0.382 − 0.923i)19-s + (−0.923 − 0.382i)21-s + (0.707 + 0.707i)23-s + (−0.923 − 0.382i)27-s + (−0.923 + 0.382i)29-s + 31-s i·33-s + (0.923 + 0.382i)37-s + (−0.707 − 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.998 + 0.0626i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.998 + 0.0626i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-0.998 + 0.0626i$
motivic weight  =  \(0\)
character  :  $\chi_{320} (43, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 320,\ (0:\ ),\ -0.998 + 0.0626i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.01876102368 + 0.5982368531i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.01876102368 + 0.5982368531i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6984296518 + 0.4010030606i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6984296518 + 0.4010030606i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−24.734341562348944392827321119237, −23.8379692384764457521714049880, −23.03021348267416328662692626239, −22.31603526953031455035871428043, −20.78851843590699678725932040331, −20.20552908153726643444969878411, −19.28203581969024246866756591596, −18.571797872564323791602811711361, −17.50548455389729880817002562494, −16.79445854440567871432802046713, −15.477518255042861223288905202302, −14.624360843976299796792213500533, −13.46985315317729201554431478084, −12.928688940601957253126321124985, −12.092267587301386794734070123355, −10.70852681921933192545892666178, −9.82325995863546476482112135081, −8.61781574687086677284527577741, −7.56528719235052473499972582889, −6.91068314826554517590923911445, −5.79257439900551979762859986607, −4.35725654479081751163894031618, −3.005002005885949295377881099910, −2.06878807341462037216618573275, −0.32993758406779597113502094196, 2.40679682580925587019540784728, 3.0656230696371206196714638611, 4.50638808978234601667273200434, 5.3314496589688984539683523749, 6.57090714881558355816176922982, 7.912821099380485703061072570895, 9.02968273608627377562164772818, 9.58442632639606840394972096203, 10.71593199713515999414242307146, 11.581715835267861124319697158003, 12.93559803938640229863826832978, 13.66135012644587102257153260403, 15.06714735032819229488614378558, 15.41964630475080736592304239726, 16.3957499062898254333288016787, 17.28717936715972765859413635944, 18.56986546494725885285437803843, 19.45317980256768989445641332975, 20.163553747518310976600190194132, 21.42367347265715844130595128429, 21.7848468598734182533525193796, 22.67025251438307283278372280512, 23.82305409408295799017974528222, 24.84294939881994833460741098766, 25.73411578908433775518480588451

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