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Properties

Label 1-837-837.41-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.998 + 0.0472i$
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  + (0.615 + 0.788i)2-s + (−0.241 + 0.970i)4-s + (0.939 + 0.342i)5-s + (0.438 + 0.898i)7-s + (−0.913 + 0.406i)8-s + (0.309 + 0.951i)10-s + (0.997 + 0.0697i)11-s + (−0.374 − 0.927i)13-s + (−0.438 + 0.898i)14-s + (−0.882 − 0.469i)16-s + (−0.913 + 0.406i)17-s + (0.309 + 0.951i)19-s + (−0.559 + 0.829i)20-s + (0.559 + 0.829i)22-s + (0.997 − 0.0697i)23-s + ⋯
L(s)  = 1  + (0.615 + 0.788i)2-s + (−0.241 + 0.970i)4-s + (0.939 + 0.342i)5-s + (0.438 + 0.898i)7-s + (−0.913 + 0.406i)8-s + (0.309 + 0.951i)10-s + (0.997 + 0.0697i)11-s + (−0.374 − 0.927i)13-s + (−0.438 + 0.898i)14-s + (−0.882 − 0.469i)16-s + (−0.913 + 0.406i)17-s + (0.309 + 0.951i)19-s + (−0.559 + 0.829i)20-s + (0.559 + 0.829i)22-s + (0.997 − 0.0697i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.998 + 0.0472i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.998 + 0.0472i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07993675232 + 3.382498585i\)
\(L(\frac12)\) \(\approx\) \(0.07993675232 + 3.382498585i\)
\(L(1)\) \(\approx\) \(1.218149309 + 1.231947688i\)
\(L(1)\) \(\approx\) \(1.218149309 + 1.231947688i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.615 + 0.788i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (0.438 + 0.898i)T \)
11 \( 1 + (0.997 + 0.0697i)T \)
13 \( 1 + (-0.374 - 0.927i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.997 - 0.0697i)T \)
29 \( 1 + (0.615 + 0.788i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.848 - 0.529i)T \)
43 \( 1 + (-0.615 - 0.788i)T \)
47 \( 1 + (-0.848 + 0.529i)T \)
53 \( 1 + (-0.913 + 0.406i)T \)
59 \( 1 + (0.374 + 0.927i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (-0.241 - 0.970i)T \)
83 \( 1 + (-0.990 + 0.139i)T \)
89 \( 1 + (0.104 + 0.994i)T \)
97 \( 1 + (0.438 + 0.898i)T \)
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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

−21.46695174068356870320294750503, −20.91420639691748162602737139442, −19.94761843885092918739637234430, −19.57573115882638261763366646218, −18.40908426855819643729163156378, −17.48884769542072866992022339704, −17.004394794435762704778622162814, −15.84907508148689387535033104125, −14.65733324152461413245463257591, −14.06939904140362580797678291058, −13.477587257195354614339893635114, −12.754909919760711438171901612622, −11.49893782122342805346940151055, −11.21264747970407928737127004732, −9.97506734416614194746597197385, −9.41987676009109198743259209894, −8.607733672750269364017489777094, −6.860093366670159288481132540800, −6.501481055377642487518387573788, −5.00018423534561666961837981866, −4.66174607235256536750062111799, −3.56343344965178844843681564116, −2.32191226782523352556258971991, −1.50805250919195506777749374215, −0.565960354689237400364225739124, 1.52135984768044793640573148565, 2.629810531810213609632832917816, 3.51771082026682816623337694064, 4.86990959433510492093047782389, 5.46461959516152382560080464731, 6.36590114833483814974352832268, 6.99736581089458907044345784479, 8.2785328882301510825829034905, 8.88355328892832523013977527357, 9.83965429704259023456217373922, 10.98153885358306873620418071816, 12.01767287815993414419512451095, 12.68601083173962126418185643516, 13.57761394255204057915187510452, 14.44130629215001294432862491954, 14.93307724908940843823158422376, 15.66873284354844674345227987073, 16.87043950911130317453841722948, 17.41234048768416826554959987698, 18.06983544633897903876964853728, 18.89626249086828629314746299379, 20.20547884401086056501991213469, 21.00709265675165957586191334149, 21.86599262620492756751896677359, 22.25101278919913191999935566664

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