Properties

Label 1-837-837.344-r0-0-0
Degree $1$
Conductor $837$
Sign $0.0610 + 0.998i$
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 + (−0.766 − 0.642i)5-s + (0.559 + 0.829i)7-s + (−0.913 − 0.406i)8-s + (0.309 − 0.951i)10-s + (0.438 − 0.898i)11-s + (−0.990 + 0.139i)13-s + (−0.559 + 0.829i)14-s + (0.0348 − 0.999i)16-s + (0.913 + 0.406i)17-s + (0.309 − 0.951i)19-s + (0.997 − 0.0697i)20-s + (0.997 + 0.0697i)22-s + (0.438 + 0.898i)23-s + ⋯
L(s)  = 1  + (0.374 + 0.927i)2-s + (−0.719 + 0.694i)4-s + (−0.766 − 0.642i)5-s + (0.559 + 0.829i)7-s + (−0.913 − 0.406i)8-s + (0.309 − 0.951i)10-s + (0.438 − 0.898i)11-s + (−0.990 + 0.139i)13-s + (−0.559 + 0.829i)14-s + (0.0348 − 0.999i)16-s + (0.913 + 0.406i)17-s + (0.309 − 0.951i)19-s + (0.997 − 0.0697i)20-s + (0.997 + 0.0697i)22-s + (0.438 + 0.898i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.0610 + 0.998i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.0610 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.033802130 + 0.9724781723i\)
\(L(\frac12)\) \(\approx\) \(1.033802130 + 0.9724781723i\)
\(L(1)\) \(\approx\) \(0.9790641572 + 0.5286221327i\)
\(L(1)\) \(\approx\) \(0.9790641572 + 0.5286221327i\)

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 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (0.559 + 0.829i)T \)
11 \( 1 + (0.438 - 0.898i)T \)
13 \( 1 + (-0.990 + 0.139i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.438 + 0.898i)T \)
29 \( 1 + (-0.374 - 0.927i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.882 + 0.469i)T \)
43 \( 1 + (0.374 + 0.927i)T \)
47 \( 1 + (0.882 - 0.469i)T \)
53 \( 1 + (0.913 + 0.406i)T \)
59 \( 1 + (-0.990 + 0.139i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.913 + 0.406i)T \)
79 \( 1 + (0.719 + 0.694i)T \)
83 \( 1 + (-0.615 + 0.788i)T \)
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

−22.08415390492961109668550995974, −20.9642446306917923699504965268, −20.33368325713774709324168936544, −19.74333084755202983265543300267, −18.9064699362617420878470977414, −18.186908139163067385322962887510, −17.33025045504887981344239446245, −16.38465860380754257118512268572, −15.07897154832867384918982632952, −14.44214510566158165694861442284, −14.144018077749249940333748228782, −12.66167296109067966976161357121, −12.18834499253125273135630460261, −11.36595555101930735944794348599, −10.472753892103188001054352658198, −10.00478545627329866315064466915, −8.863891987671352457320273020637, −7.57529441487117884313226112581, −7.15294199482220792160097545734, −5.69188167990749286728931115525, −4.61460903334772926622139942097, −4.013195413128610742539843989577, −3.060761014614058946713884518538, −2.01144298993667872344936218294, −0.77712804264955819191695342342, 0.965621032543672070748589826855, 2.72915950136793737725527402306, 3.7460226786557177434855720288, 4.7463512211333526368407683510, 5.38099619625546210255300493765, 6.27279248745997742801517235465, 7.56935160702090331786670141283, 7.97286125028575659089380602871, 8.98268457607693734002811592989, 9.49519356398554708062214621249, 11.27632859108879892788893575631, 11.87221067204643163143140852004, 12.58980390006026166085874177549, 13.528137645879425934573178673367, 14.44847309898975085038195938816, 15.210411340144884792452556020233, 15.716887180177797251426301087861, 16.83377272235679221616205865352, 17.09831653045024156817518235985, 18.273767232360642222636451228373, 19.09699419787830650129692223746, 19.80725422495404770851086705135, 21.14222842540404764570419489529, 21.5601606249892279666209885512, 22.3817086825629079843885259319

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