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Properties

Label 1-837-837.725-r0-0-0
Degree $1$
Conductor $837$
Sign $0.182 + 0.983i$
Analytic cond. $3.88701$
Root an. cond. $3.88701$
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  + (0.374 − 0.927i)2-s + (−0.719 − 0.694i)4-s + (−0.173 − 0.984i)5-s + (−0.997 − 0.0697i)7-s + (−0.913 + 0.406i)8-s + (−0.978 − 0.207i)10-s + (−0.997 − 0.0697i)11-s + (0.615 − 0.788i)13-s + (−0.438 + 0.898i)14-s + (0.0348 + 0.999i)16-s + (−0.809 − 0.587i)17-s + (0.669 − 0.743i)19-s + (−0.559 + 0.829i)20-s + (−0.438 + 0.898i)22-s + (0.438 − 0.898i)23-s + ⋯
L(s)  = 1  + (0.374 − 0.927i)2-s + (−0.719 − 0.694i)4-s + (−0.173 − 0.984i)5-s + (−0.997 − 0.0697i)7-s + (−0.913 + 0.406i)8-s + (−0.978 − 0.207i)10-s + (−0.997 − 0.0697i)11-s + (0.615 − 0.788i)13-s + (−0.438 + 0.898i)14-s + (0.0348 + 0.999i)16-s + (−0.809 − 0.587i)17-s + (0.669 − 0.743i)19-s + (−0.559 + 0.829i)20-s + (−0.438 + 0.898i)22-s + (0.438 − 0.898i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.182 + 0.983i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (725, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.182 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2550687574 - 0.2120143332i\)
\(L(\frac12)\) \(\approx\) \(-0.2550687574 - 0.2120143332i\)
\(L(1)\) \(\approx\) \(0.5168961922 - 0.5772298877i\)
\(L(1)\) \(\approx\) \(0.5168961922 - 0.5772298877i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.374 - 0.927i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (-0.997 - 0.0697i)T \)
11 \( 1 + (-0.997 - 0.0697i)T \)
13 \( 1 + (0.615 - 0.788i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.438 - 0.898i)T \)
29 \( 1 + (-0.374 + 0.927i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.0348 + 0.999i)T \)
43 \( 1 + (-0.990 + 0.139i)T \)
47 \( 1 + (0.882 + 0.469i)T \)
53 \( 1 + (-0.104 + 0.994i)T \)
59 \( 1 + (0.374 + 0.927i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.961 - 0.275i)T \)
83 \( 1 + (0.990 - 0.139i)T \)
89 \( 1 + (-0.104 - 0.994i)T \)
97 \( 1 + (0.559 - 0.829i)T \)
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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

−22.90350803849840160971616448434, −22.16340173670660136927665349697, −21.484635769617383660014479312194, −20.55806348336107596817198973784, −19.15886510473765876481478513862, −18.78161300924681508065425667277, −17.92889535518600049259242299714, −17.07230708252851463465072665486, −16.00589614515711021964252099142, −15.621880109617180061724479890491, −14.88834394712329152240973100244, −13.730931905587712430990582425519, −13.431975082834943309346974494049, −12.36944835011501929044927713181, −11.42526890946932087344921523656, −10.347540861784304001632536554867, −9.52430681163159950515710365008, −8.52467502960649096932754749029, −7.5484044745297687396174218075, −6.826292599875932082359736052136, −6.124548084699227067250757960618, −5.27491408855914287646881464345, −3.8617509148699623123236200847, −3.40518374229947704206864716934, −2.20802862809123357604966001274, 0.14068800547654770895358664573, 1.18805888861956793835895326242, 2.65444555037224591738006091305, 3.28952460389326927819440882842, 4.48271224968218770757159669128, 5.19480830936274992158557915553, 6.05236991512010871112951147856, 7.33351836133782278041707193512, 8.64482503878505297948014135945, 9.10096228585419029656783735452, 10.15002587033494971410200031992, 10.833422911678266593303644021261, 11.83267173887303064791826953728, 12.73400066915096321402451237874, 13.194046149403885781988113616571, 13.73069643104004169959605100642, 15.20179182887787049012945167560, 15.79681953533746485717282553234, 16.58594524322070033216199833228, 17.81745555523834724118801128649, 18.41635488519650996400056620512, 19.36706036030847464428232784924, 20.25983819217316075311100360461, 20.42453895929265022000403562186, 21.42782989783892286404102599585

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