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Properties

Label 1-837-837.536-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.262 + 0.964i$
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  + (0.241 + 0.970i)2-s + (−0.882 + 0.469i)4-s + (−0.766 + 0.642i)5-s + (−0.615 − 0.788i)7-s + (−0.669 − 0.743i)8-s + (−0.809 − 0.587i)10-s + (−0.990 + 0.139i)11-s + (−0.719 − 0.694i)13-s + (0.615 − 0.788i)14-s + (0.559 − 0.829i)16-s + (−0.669 − 0.743i)17-s + (−0.809 − 0.587i)19-s + (0.374 − 0.927i)20-s + (−0.374 − 0.927i)22-s + (−0.990 − 0.139i)23-s + ⋯
L(s)  = 1  + (0.241 + 0.970i)2-s + (−0.882 + 0.469i)4-s + (−0.766 + 0.642i)5-s + (−0.615 − 0.788i)7-s + (−0.669 − 0.743i)8-s + (−0.809 − 0.587i)10-s + (−0.990 + 0.139i)11-s + (−0.719 − 0.694i)13-s + (0.615 − 0.788i)14-s + (0.559 − 0.829i)16-s + (−0.669 − 0.743i)17-s + (−0.809 − 0.587i)19-s + (0.374 − 0.927i)20-s + (−0.374 − 0.927i)22-s + (−0.990 − 0.139i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.262 + 0.964i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (536, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.262 + 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1387163451 + 0.1814533377i\)
\(L(\frac12)\) \(\approx\) \(0.1387163451 + 0.1814533377i\)
\(L(1)\) \(\approx\) \(0.5191099077 + 0.2291382543i\)
\(L(1)\) \(\approx\) \(0.5191099077 + 0.2291382543i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.241 + 0.970i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (-0.615 - 0.788i)T \)
11 \( 1 + (-0.990 + 0.139i)T \)
13 \( 1 + (-0.719 - 0.694i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.990 - 0.139i)T \)
29 \( 1 + (0.241 + 0.970i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.438 + 0.898i)T \)
43 \( 1 + (-0.241 - 0.970i)T \)
47 \( 1 + (-0.438 - 0.898i)T \)
53 \( 1 + (-0.669 - 0.743i)T \)
59 \( 1 + (0.719 + 0.694i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.669 - 0.743i)T \)
79 \( 1 + (-0.882 - 0.469i)T \)
83 \( 1 + (-0.961 - 0.275i)T \)
89 \( 1 + (0.978 + 0.207i)T \)
97 \( 1 + (-0.615 - 0.788i)T \)
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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

−21.513498926712266223134254099017, −21.04803924317118346747243004383, −20.03544677234660045290227343788, −19.27899897487179246277887447241, −18.96997882818636597792746696806, −17.92295056142754562368961045126, −16.95964025902715147685458779428, −15.90665684977360248985003600736, −15.30569886508057050692043893277, −14.34052824316088035513919463291, −13.2358374711876004827891271547, −12.59075349205311478764889121311, −12.07938404256598935539804586624, −11.20515486920656992254588502720, −10.24975287227484070078428589555, −9.40258353562467612839941086151, −8.56965363049224379060511364837, −7.87422290611905759544925569831, −6.33878870088853648404168249926, −5.43739701683218102333094273049, −4.4778028040147351510911249651, −3.756876931957298157303707236026, −2.58229276745372651419955233506, −1.80257548523456608516303770987, −0.141578364597019484027161619009, 0.311205439077772727953435584529, 2.633371681816185498920964572552, 3.4411768485943123700398909857, 4.44953146644809099578751465570, 5.21062870222921697974979703651, 6.57169902164774729624131423683, 7.02763178181799952001937217821, 7.835285397097392323035619958051, 8.59407333435707307205011543774, 9.946124560049984496802649318918, 10.45564581726298277812949376240, 11.67186102162374013130093797757, 12.7154627878781108169262766305, 13.31761160126178276917467445131, 14.225905066417500523609199295513, 15.076445193417204863778674699, 15.72780899782621222556183057839, 16.308327917569255770659898384576, 17.33161306509076358595725402910, 18.05888588090251460791290962228, 18.81756391982406354651779094199, 19.78641586090924413001982254007, 20.441562037616451367871151756189, 21.8405987073673115867135810427, 22.32429318122964298260717014762

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