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Properties

Label 1-805-805.179-r0-0-0
Degree $1$
Conductor $805$
Sign $-0.451 - 0.892i$
Analytic cond. $3.73840$
Root an. cond. $3.73840$
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  + (0.888 − 0.458i)2-s + (0.786 − 0.618i)3-s + (0.580 − 0.814i)4-s + (0.415 − 0.909i)6-s + (0.142 − 0.989i)8-s + (0.235 − 0.971i)9-s + (−0.888 − 0.458i)11-s + (−0.0475 − 0.998i)12-s + (0.654 + 0.755i)13-s + (−0.327 − 0.945i)16-s + (0.995 − 0.0950i)17-s + (−0.235 − 0.971i)18-s + (−0.995 − 0.0950i)19-s − 22-s + (−0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.888 − 0.458i)2-s + (0.786 − 0.618i)3-s + (0.580 − 0.814i)4-s + (0.415 − 0.909i)6-s + (0.142 − 0.989i)8-s + (0.235 − 0.971i)9-s + (−0.888 − 0.458i)11-s + (−0.0475 − 0.998i)12-s + (0.654 + 0.755i)13-s + (−0.327 − 0.945i)16-s + (0.995 − 0.0950i)17-s + (−0.235 − 0.971i)18-s + (−0.995 − 0.0950i)19-s − 22-s + (−0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $-0.451 - 0.892i$
Analytic conductor: \(3.73840\)
Root analytic conductor: \(3.73840\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{805} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 805,\ (0:\ ),\ -0.451 - 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.671601804 - 2.720059973i\)
\(L(\frac12)\) \(\approx\) \(1.671601804 - 2.720059973i\)
\(L(1)\) \(\approx\) \(1.764356806 - 1.280318382i\)
\(L(1)\) \(\approx\) \(1.764356806 - 1.280318382i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.888 - 0.458i)T \)
3 \( 1 + (0.786 - 0.618i)T \)
11 \( 1 + (-0.888 - 0.458i)T \)
13 \( 1 + (0.654 + 0.755i)T \)
17 \( 1 + (0.995 - 0.0950i)T \)
19 \( 1 + (-0.995 - 0.0950i)T \)
29 \( 1 + (0.415 - 0.909i)T \)
31 \( 1 + (0.928 - 0.371i)T \)
37 \( 1 + (-0.235 + 0.971i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.981 - 0.189i)T \)
59 \( 1 + (-0.327 + 0.945i)T \)
61 \( 1 + (-0.786 - 0.618i)T \)
67 \( 1 + (-0.0475 + 0.998i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (-0.580 + 0.814i)T \)
79 \( 1 + (0.981 - 0.189i)T \)
83 \( 1 + (0.959 - 0.281i)T \)
89 \( 1 + (0.928 + 0.371i)T \)
97 \( 1 + (0.959 + 0.281i)T \)
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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

−22.5456798147860145555920079246, −21.58764447391081803808584450301, −20.9762106919215107530177565243, −20.48663684374000327158585454596, −19.57517270341452935035583003270, −18.544091654617879764374474848971, −17.47508205717654158214523464355, −16.55648881281393204723190253238, −15.72045099430417104181389145490, −15.286270413978951828213111085134, −14.42878522124165486737927101043, −13.72502934802636261726137853673, −12.87291998511456227333615020471, −12.23557755946420094776680271590, −10.74091749961866514760414181442, −10.4119672215701402592582132332, −9.035082157452032530882061260183, −8.108664340485099596002844275446, −7.62808658227871039614595750231, −6.39027465464612043690967101374, −5.329249205689324077049342976301, −4.66235979057973457782597370549, −3.57875291622337576972559687956, −2.94230268589894819331333798749, −1.87378293528492821669750159400, 1.00062876790331453381424506988, 2.080586737174760710498553714021, 2.918994336192004863413635455676, 3.76866412693070164178953570564, 4.769467695792948580118490548741, 6.00976683684087248933794154448, 6.61456553650990714986422722804, 7.75997726232500931718982113232, 8.55566509733424620622636785944, 9.70126309652334149391854930670, 10.51184093726937845826123803747, 11.59511076718309810035689011021, 12.25926522767197009097408550429, 13.283661626881646302679842901561, 13.616529234874158016907276141435, 14.477400433197083101023154668961, 15.286718778926423697398345011, 16.01317936019944219448528047545, 17.149721736336366661619913387217, 18.58295370542560842473952872872, 18.82120618691180879534751181051, 19.59677354778961918876223485870, 20.67130530361742936199299807860, 21.04234641295054500522058227429, 21.72725791121806011971299260262

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