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Properties

Label 1-837-837.410-r1-0-0
Degree $1$
Conductor $837$
Sign $0.782 - 0.623i$
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  + (−0.438 − 0.898i)2-s + (−0.615 + 0.788i)4-s + (0.939 + 0.342i)5-s + (−0.882 − 0.469i)7-s + (0.978 + 0.207i)8-s + (−0.104 − 0.994i)10-s + (0.882 + 0.469i)11-s + (−0.997 + 0.0697i)13-s + (−0.0348 + 0.999i)14-s + (−0.241 − 0.970i)16-s + (−0.309 − 0.951i)17-s + (0.913 + 0.406i)19-s + (−0.848 + 0.529i)20-s + (0.0348 − 0.999i)22-s + (−0.0348 + 0.999i)23-s + ⋯
L(s)  = 1  + (−0.438 − 0.898i)2-s + (−0.615 + 0.788i)4-s + (0.939 + 0.342i)5-s + (−0.882 − 0.469i)7-s + (0.978 + 0.207i)8-s + (−0.104 − 0.994i)10-s + (0.882 + 0.469i)11-s + (−0.997 + 0.0697i)13-s + (−0.0348 + 0.999i)14-s + (−0.241 − 0.970i)16-s + (−0.309 − 0.951i)17-s + (0.913 + 0.406i)19-s + (−0.848 + 0.529i)20-s + (0.0348 − 0.999i)22-s + (−0.0348 + 0.999i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.782 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.782 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.782 - 0.623i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (410, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.782 - 0.623i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.512620836 - 0.5288143389i\)
\(L(\frac12)\) \(\approx\) \(1.512620836 - 0.5288143389i\)
\(L(1)\) \(\approx\) \(0.8748791745 - 0.2844618124i\)
\(L(1)\) \(\approx\) \(0.8748791745 - 0.2844618124i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.438 - 0.898i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (-0.882 - 0.469i)T \)
11 \( 1 + (0.882 + 0.469i)T \)
13 \( 1 + (-0.997 + 0.0697i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.0348 + 0.999i)T \)
29 \( 1 + (-0.438 - 0.898i)T \)
37 \( 1 + T \)
41 \( 1 + (0.241 - 0.970i)T \)
43 \( 1 + (0.559 - 0.829i)T \)
47 \( 1 + (-0.961 - 0.275i)T \)
53 \( 1 + (-0.669 + 0.743i)T \)
59 \( 1 + (-0.438 + 0.898i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.374 + 0.927i)T \)
83 \( 1 + (-0.559 + 0.829i)T \)
89 \( 1 + (-0.669 - 0.743i)T \)
97 \( 1 + (0.848 - 0.529i)T \)
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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

−22.101946903424711398768312235948, −21.607840429933479515024474927766, −20.06428886664125093310240287913, −19.61892531899433278486908810719, −18.64861590554139596573242120305, −17.92023272543742234782736129426, −17.04955869719362932172098263241, −16.565624284556011074006595260902, −15.79256304695751582207937419058, −14.642493834012279751697683716281, −14.27167773045836930746767379022, −13.06773302917582996389733822930, −12.64955997294898351324283383142, −11.241605062570416136709208964, −10.091361954230755322501209792442, −9.46881831328959530534522970482, −8.94124348406638383090806947755, −7.9437893184557152810426261977, −6.6630131678345734567113658793, −6.27515799182630924221194435678, −5.3626464495292016491678326022, −4.460133240671329829990078330526, −3.0311610085080600682955985960, −1.77249209665863754180896007242, −0.633495204043819446479599996084, 0.68921744109176807307613456967, 1.83241595761500982446740059151, 2.73521678131901387158007015586, 3.6241296587170731782900354121, 4.67948804312298635564762498341, 5.83558591829575723997067824653, 7.03796194674059701117796400916, 7.54742851985728819864405262468, 9.21999057610776532131598677620, 9.551686983196517275279056908284, 10.091024014800711745928793402225, 11.18046951433944561530501964374, 12.01599119805463834076165207887, 12.81092118788555801370808014025, 13.769793252932469487869368736363, 14.12647401251089800469827205192, 15.466603867820176625885388738640, 16.70805719133754433809931743879, 17.11825589106944265241725790186, 17.94117369767471505387572112476, 18.69963503067303057643985145785, 19.629131793326701577030818583138, 20.101644159140185413543771067519, 20.99803183607279488381181635718, 21.87317049737290637457605280953

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