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Properties

Label 1-320-320.67-r0-0-0
Degree $1$
Conductor $320$
Sign $-0.998 - 0.0626i$
Analytic cond. $1.48607$
Root an. cond. $1.48607$
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  + (0.382 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.998 - 0.0626i$
Analytic conductor: \(1.48607\)
Root analytic conductor: \(1.48607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 320,\ (0:\ ),\ -0.998 - 0.0626i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01876102368 - 0.5982368531i\)
\(L(\frac12)\) \(\approx\) \(0.01876102368 - 0.5982368531i\)
\(L(1)\) \(\approx\) \(0.6984296518 - 0.4010030606i\)
\(L(1)\) \(\approx\) \(0.6984296518 - 0.4010030606i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.382 + 0.923i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (-0.923 - 0.382i)T \)
31 \( 1 + T \)
37 \( 1 + (0.923 - 0.382i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (-0.382 - 0.923i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.382 - 0.923i)T \)
59 \( 1 + (0.382 + 0.923i)T \)
61 \( 1 + (-0.923 - 0.382i)T \)
67 \( 1 + (-0.382 + 0.923i)T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.923 - 0.382i)T \)
89 \( 1 + (0.707 - 0.707i)T \)
97 \( 1 - iT \)
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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

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

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