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Properties

Label 1-837-837.193-r0-0-0
Degree $1$
Conductor $837$
Sign $0.623 + 0.782i$
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  + (−0.997 − 0.0697i)2-s + (0.990 + 0.139i)4-s + (0.173 + 0.984i)5-s + (0.848 − 0.529i)7-s + (−0.978 − 0.207i)8-s + (−0.104 − 0.994i)10-s + (0.848 − 0.529i)11-s + (0.559 + 0.829i)13-s + (−0.882 + 0.469i)14-s + (0.961 + 0.275i)16-s + (0.309 + 0.951i)17-s + (0.913 + 0.406i)19-s + (0.0348 + 0.999i)20-s + (−0.882 + 0.469i)22-s + (−0.882 + 0.469i)23-s + ⋯
L(s)  = 1  + (−0.997 − 0.0697i)2-s + (0.990 + 0.139i)4-s + (0.173 + 0.984i)5-s + (0.848 − 0.529i)7-s + (−0.978 − 0.207i)8-s + (−0.104 − 0.994i)10-s + (0.848 − 0.529i)11-s + (0.559 + 0.829i)13-s + (−0.882 + 0.469i)14-s + (0.961 + 0.275i)16-s + (0.309 + 0.951i)17-s + (0.913 + 0.406i)19-s + (0.0348 + 0.999i)20-s + (−0.882 + 0.469i)22-s + (−0.882 + 0.469i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.623 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.623 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.623 + 0.782i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.623 + 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.054977620 + 0.5084138110i\)
\(L(\frac12)\) \(\approx\) \(1.054977620 + 0.5084138110i\)
\(L(1)\) \(\approx\) \(0.8658918864 + 0.1670515694i\)
\(L(1)\) \(\approx\) \(0.8658918864 + 0.1670515694i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.997 - 0.0697i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (0.848 - 0.529i)T \)
11 \( 1 + (0.848 - 0.529i)T \)
13 \( 1 + (0.559 + 0.829i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.882 + 0.469i)T \)
29 \( 1 + (-0.997 - 0.0697i)T \)
37 \( 1 + T \)
41 \( 1 + (0.961 - 0.275i)T \)
43 \( 1 + (0.438 + 0.898i)T \)
47 \( 1 + (-0.719 + 0.694i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (-0.997 + 0.0697i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.615 - 0.788i)T \)
83 \( 1 + (0.438 + 0.898i)T \)
89 \( 1 + (0.669 + 0.743i)T \)
97 \( 1 + (0.0348 + 0.999i)T \)
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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

−21.81824127693770454477545481652, −20.88856851322073116701377775973, −20.27122869102281911481810782072, −19.89738895998413912416543379066, −18.540602958836483385682931989987, −18.01374294059996088942207213156, −17.3675557145093454941133024748, −16.49103801344732961825932762701, −15.82441387114124005227782706238, −14.97324019595240499908865597873, −14.09173129359306568821353446085, −12.90341210151146935196061965572, −11.8980682656637268826283693738, −11.55817153901651358092799292956, −10.3541381375495019846531813681, −9.38258939192394575730419741654, −8.923515355911918359679814236648, −7.98262566968367940202869802278, −7.3343994825111602050409838188, −5.96781316307366195380601080277, −5.35368334923902502255233286159, −4.19609181699789043059578881193, −2.735904181031489448128075328740, −1.66829670726934624294574818300, −0.8519289551696091888735999531, 1.27928230912451033099006075425, 1.9459106129142912953935958004, 3.336396687951348746661631621160, 4.04862718799226135647024672684, 5.85842586739230571277791149286, 6.39014868635985524246297468327, 7.50777769774980736951937316972, 7.98547445864521321612670766804, 9.15492267741240880402890107038, 9.86942780133060063258664913915, 10.90280776579417599712250631693, 11.28430134740740602931725230979, 12.05606904217043509011688497267, 13.54364659513079345831645363590, 14.40332396280061307026059113967, 14.87862498527485674341172172624, 16.12276136762111637864365589945, 16.73913365091081039918920567445, 17.6675019049100160942007526018, 18.16525293581849483280894227530, 19.04534006084395133167358698515, 19.61871103261709814813295487118, 20.60231481943036023950040661458, 21.376131442443005770400281004587, 21.992909471940646201231261018955

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