Properties

Label 1-837-837.562-r0-0-0
Degree $1$
Conductor $837$
Sign $0.855 + 0.517i$
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  + (0.438 + 0.898i)2-s + (−0.615 + 0.788i)4-s + (0.766 − 0.642i)5-s + (0.0348 + 0.999i)7-s + (−0.978 − 0.207i)8-s + (0.913 + 0.406i)10-s + (0.848 − 0.529i)11-s + (0.438 − 0.898i)13-s + (−0.882 + 0.469i)14-s + (−0.241 − 0.970i)16-s + (0.669 − 0.743i)17-s + (−0.104 − 0.994i)19-s + (0.0348 + 0.999i)20-s + (0.848 + 0.529i)22-s + (0.0348 − 0.999i)23-s + ⋯
L(s)  = 1  + (0.438 + 0.898i)2-s + (−0.615 + 0.788i)4-s + (0.766 − 0.642i)5-s + (0.0348 + 0.999i)7-s + (−0.978 − 0.207i)8-s + (0.913 + 0.406i)10-s + (0.848 − 0.529i)11-s + (0.438 − 0.898i)13-s + (−0.882 + 0.469i)14-s + (−0.241 − 0.970i)16-s + (0.669 − 0.743i)17-s + (−0.104 − 0.994i)19-s + (0.0348 + 0.999i)20-s + (0.848 + 0.529i)22-s + (0.0348 − 0.999i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.855 + 0.517i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (562, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.855 + 0.517i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.978850852 + 0.5519700605i\)
\(L(\frac12)\) \(\approx\) \(1.978850852 + 0.5519700605i\)
\(L(1)\) \(\approx\) \(1.393675524 + 0.4773486369i\)
\(L(1)\) \(\approx\) \(1.393675524 + 0.4773486369i\)

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 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 + (0.0348 + 0.999i)T \)
11 \( 1 + (0.848 - 0.529i)T \)
13 \( 1 + (0.438 - 0.898i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.0348 - 0.999i)T \)
29 \( 1 + (0.438 + 0.898i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.719 - 0.694i)T \)
43 \( 1 + (-0.997 - 0.0697i)T \)
47 \( 1 + (0.961 + 0.275i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.997 + 0.0697i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (0.669 + 0.743i)T \)
79 \( 1 + (0.990 - 0.139i)T \)
83 \( 1 + (0.438 + 0.898i)T \)
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

−21.87459112698334123650377726193, −21.32743921400284578624427653727, −20.60944814116521567824418251451, −19.73661677121109215340188663398, −19.00673849247609787787301095796, −18.290737308840448101358543451434, −17.25938622557354193331182765209, −16.82282389675651856636368373228, −15.25706895130974678174396123303, −14.49866339178840220648105600937, −13.84966481732971067188011516162, −13.35316159409539332099409365540, −12.17414693562405920551497338655, −11.48113148946979147591423626983, −10.507669110525939559339100694280, −9.96420116189666602495038483144, −9.27198587584291829927753180082, −7.97390076285628901251843434366, −6.67527919613005709755654513182, −6.15538442020855874866799645229, −4.94637591804208788901880059852, −3.88150466769927781054066688589, −3.3665698579789074369414585257, −1.83151373539775800398255807347, −1.41748663789066736250781252212, 0.89552559166529373056454803925, 2.48952687297139799936136218869, 3.41444654759028362940831823974, 4.74437626170641871976330711080, 5.41489961158112362370304708326, 6.09367215853744206871070959169, 6.948959763111439520928571566504, 8.24945605640499381169328200384, 8.867291714053652785121847728479, 9.40304439591285734274141540652, 10.72631883709506071751789058641, 12.08664253690511290389860078281, 12.461233358370267944082758649966, 13.50029949669781341835789365173, 14.085277904896082648684077119732, 14.99197668695962388387295559083, 15.80806428714567156505260730083, 16.52147624762250261768107231543, 17.2682016324178811769714925451, 18.07911600190050700524488882175, 18.66493128137500596420695708904, 19.9886808429208524270617493982, 20.8680091997908786980657415581, 21.66657884336322462349896843483, 22.14977981779583319582356654966

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