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Properties

Label 1-837-837.238-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.998 - 0.0610i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
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  + (0.990 − 0.139i)2-s + (0.961 − 0.275i)4-s + (0.173 + 0.984i)5-s + (−0.997 − 0.0697i)7-s + (0.913 − 0.406i)8-s + (0.309 + 0.951i)10-s + (−0.559 + 0.829i)11-s + (0.615 − 0.788i)13-s + (−0.997 + 0.0697i)14-s + (0.848 − 0.529i)16-s + (−0.913 + 0.406i)17-s + (0.309 + 0.951i)19-s + (0.438 + 0.898i)20-s + (−0.438 + 0.898i)22-s + (−0.559 − 0.829i)23-s + ⋯
L(s)  = 1  + (0.990 − 0.139i)2-s + (0.961 − 0.275i)4-s + (0.173 + 0.984i)5-s + (−0.997 − 0.0697i)7-s + (0.913 − 0.406i)8-s + (0.309 + 0.951i)10-s + (−0.559 + 0.829i)11-s + (0.615 − 0.788i)13-s + (−0.997 + 0.0697i)14-s + (0.848 − 0.529i)16-s + (−0.913 + 0.406i)17-s + (0.309 + 0.951i)19-s + (0.438 + 0.898i)20-s + (−0.438 + 0.898i)22-s + (−0.559 − 0.829i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.998 - 0.0610i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (238, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.998 - 0.0610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01458806790 + 0.4772645793i\)
\(L(\frac12)\) \(\approx\) \(0.01458806790 + 0.4772645793i\)
\(L(1)\) \(\approx\) \(1.432590751 + 0.1599838462i\)
\(L(1)\) \(\approx\) \(1.432590751 + 0.1599838462i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.990 - 0.139i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (-0.997 - 0.0697i)T \)
11 \( 1 + (-0.559 + 0.829i)T \)
13 \( 1 + (0.615 - 0.788i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.559 - 0.829i)T \)
29 \( 1 + (-0.990 + 0.139i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.0348 - 0.999i)T \)
43 \( 1 + (-0.990 + 0.139i)T \)
47 \( 1 + (0.0348 + 0.999i)T \)
53 \( 1 + (-0.913 + 0.406i)T \)
59 \( 1 + (-0.615 + 0.788i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.913 - 0.406i)T \)
79 \( 1 + (-0.961 - 0.275i)T \)
83 \( 1 + (0.374 - 0.927i)T \)
89 \( 1 + (0.104 + 0.994i)T \)
97 \( 1 + (-0.997 - 0.0697i)T \)
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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

−21.83643402999482558826591316642, −20.81070936854780356683064559250, −20.12234632504925685241505786360, −19.446339940032589692393847636620, −18.412033712289390936961712440047, −17.21356688957157687182503495147, −16.34836316869562571550428449176, −15.980535454066649173357795213575, −15.2607667271834920703474070417, −13.879378307363364915560621568756, −13.25499729524583284765795319373, −13.051399978488594670766599258257, −11.704532865358812735278694827411, −11.31973668070584181221444513823, −9.97620553285854121640364461963, −9.073617107852620343489849706632, −8.20951146427450967363656983044, −7.03429864708471869232969773363, −6.2017439283360757767756378790, −5.455948370886047863379182055857, −4.53665020893138328050094486954, −3.61771483632108332079879602066, −2.66919727470049597348092451624, −1.49733331037585623370601864573, −0.06292241274704529993323431682, 1.77965455130521028406933388789, 2.69105542211913037334220815135, 3.48858782301111439475871833975, 4.317056916261168109810535589141, 5.70455308859953349573716475138, 6.202921344439254933002684178986, 7.1057838041345098732101694632, 7.87345808917156278250839211074, 9.44444113388374944325451023619, 10.47812527473338478111203253523, 10.65366385938860455332882288712, 11.92229328803481075547227242939, 12.82013387300472766040908117752, 13.289376867080714342477073099401, 14.27693336869003104996607923353, 15.04292951292508070719699558736, 15.68897203114510738181789642129, 16.41123884248725089537396206886, 17.6467030452309148823481414932, 18.48443532496652644526571937750, 19.252502208615620098948794290158, 20.22659534857176609978368343544, 20.6871681556062401603340824243, 21.86772833245231978780303635118, 22.44551169980825639092444000985

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