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Properties

Label 1-837-837.338-r1-0-0
Degree $1$
Conductor $837$
Sign $0.986 + 0.162i$
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  + (−0.0348 − 0.999i)2-s + (−0.997 + 0.0697i)4-s + (−0.766 + 0.642i)5-s + (0.961 + 0.275i)7-s + (0.104 + 0.994i)8-s + (0.669 + 0.743i)10-s + (−0.961 − 0.275i)11-s + (−0.882 + 0.469i)13-s + (0.241 − 0.970i)14-s + (0.990 − 0.139i)16-s + (0.809 − 0.587i)17-s + (−0.978 + 0.207i)19-s + (0.719 − 0.694i)20-s + (−0.241 + 0.970i)22-s + (0.241 − 0.970i)23-s + ⋯
L(s)  = 1  + (−0.0348 − 0.999i)2-s + (−0.997 + 0.0697i)4-s + (−0.766 + 0.642i)5-s + (0.961 + 0.275i)7-s + (0.104 + 0.994i)8-s + (0.669 + 0.743i)10-s + (−0.961 − 0.275i)11-s + (−0.882 + 0.469i)13-s + (0.241 − 0.970i)14-s + (0.990 − 0.139i)16-s + (0.809 − 0.587i)17-s + (−0.978 + 0.207i)19-s + (0.719 − 0.694i)20-s + (−0.241 + 0.970i)22-s + (0.241 − 0.970i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.986 + 0.162i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (338, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.986 + 0.162i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.018600787 + 0.08351373040i\)
\(L(\frac12)\) \(\approx\) \(1.018600787 + 0.08351373040i\)
\(L(1)\) \(\approx\) \(0.7538965551 - 0.2452937126i\)
\(L(1)\) \(\approx\) \(0.7538965551 - 0.2452937126i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.0348 - 0.999i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (0.961 + 0.275i)T \)
11 \( 1 + (-0.961 - 0.275i)T \)
13 \( 1 + (-0.882 + 0.469i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (0.241 - 0.970i)T \)
29 \( 1 + (-0.0348 - 0.999i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.990 - 0.139i)T \)
43 \( 1 + (0.848 - 0.529i)T \)
47 \( 1 + (0.374 - 0.927i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (-0.0348 + 0.999i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.104 + 0.994i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.438 + 0.898i)T \)
83 \( 1 + (-0.848 + 0.529i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (-0.719 + 0.694i)T \)
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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

−21.95876085633536058918344024258, −21.21204754825942695618860486560, −20.31774666507038912677502098825, −19.41490199323116906275046705981, −18.633818617500251471046631199152, −17.598384896063439737791407283605, −17.138813146007556667678878160982, −16.305497591807326326002350523444, −15.359659846160330739135447359557, −14.92533645565713092494755268985, −14.065920912851273266825975341542, −12.847316919240378677914314119069, −12.552326687280625509216208394026, −11.238490962517735797714729031427, −10.3218841365578638608180606563, −9.314377739241100496170060437317, −8.24495436299223575411484614158, −7.82706282234616603701137543606, −7.17784927055942906417669921282, −5.7553096822258334692182954421, −4.94572203356167037069278452622, −4.43992765078119395367543286574, −3.24361041532065591226917471692, −1.544779461324793124237567319392, −0.35185982890789462163904205107, 0.70345744103976326842075871609, 2.254693839976377844309387364257, 2.720006664709512874582803882983, 4.01316124122484868384050141904, 4.72065594723922993081285002927, 5.670756940782593436599306972943, 7.176558719777380083702473589146, 8.02902297815633248005367332127, 8.618162892017226855141493693916, 9.913595969617070614852983019447, 10.58317488336365965372390362770, 11.39000095295383549641662151499, 11.97261549866567169108228619549, 12.76759401077238777413886789890, 13.91517084294726267005947738009, 14.611825862966897764277107214806, 15.22180607857816306401532885006, 16.49331076781022916206795136289, 17.36593817373213933666412960903, 18.39753436487277157592932273830, 18.75725408507856180031962328030, 19.4513517944030143628706128310, 20.53854684376876310851132407369, 21.05485447489577663028178510432, 21.84415817225493609936746577498

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