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Properties

Label 1-837-837.542-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.787 + 0.615i$
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  + (0.615 − 0.788i)2-s + (−0.241 − 0.970i)4-s + (−0.173 + 0.984i)5-s + (−0.997 + 0.0697i)7-s + (−0.913 − 0.406i)8-s + (0.669 + 0.743i)10-s + (0.438 − 0.898i)11-s + (0.615 + 0.788i)13-s + (−0.559 + 0.829i)14-s + (−0.882 + 0.469i)16-s + (−0.104 − 0.994i)17-s + (−0.978 + 0.207i)19-s + (0.997 − 0.0697i)20-s + (−0.438 − 0.898i)22-s + (−0.997 − 0.0697i)23-s + ⋯
L(s)  = 1  + (0.615 − 0.788i)2-s + (−0.241 − 0.970i)4-s + (−0.173 + 0.984i)5-s + (−0.997 + 0.0697i)7-s + (−0.913 − 0.406i)8-s + (0.669 + 0.743i)10-s + (0.438 − 0.898i)11-s + (0.615 + 0.788i)13-s + (−0.559 + 0.829i)14-s + (−0.882 + 0.469i)16-s + (−0.104 − 0.994i)17-s + (−0.978 + 0.207i)19-s + (0.997 − 0.0697i)20-s + (−0.438 − 0.898i)22-s + (−0.997 − 0.0697i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.787 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.787 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.787 + 0.615i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (542, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.787 + 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07304031608 - 0.2120023735i\)
\(L(\frac12)\) \(\approx\) \(-0.07304031608 - 0.2120023735i\)
\(L(1)\) \(\approx\) \(0.8335240868 - 0.3655143710i\)
\(L(1)\) \(\approx\) \(0.8335240868 - 0.3655143710i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.615 - 0.788i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (-0.997 + 0.0697i)T \)
11 \( 1 + (0.438 - 0.898i)T \)
13 \( 1 + (0.615 + 0.788i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.997 - 0.0697i)T \)
29 \( 1 + (-0.615 + 0.788i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.0348 - 0.999i)T \)
43 \( 1 + (-0.990 - 0.139i)T \)
47 \( 1 + (-0.848 - 0.529i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.990 + 0.139i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.104 + 0.994i)T \)
73 \( 1 + (0.104 - 0.994i)T \)
79 \( 1 + (-0.961 + 0.275i)T \)
83 \( 1 + (-0.615 + 0.788i)T \)
89 \( 1 + (-0.104 + 0.994i)T \)
97 \( 1 + (0.438 - 0.898i)T \)
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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

−22.84184946660661082129699414036, −21.9722636020480647930420498914, −21.17808564991114914235585994250, −20.200480436103982770098114448735, −19.741897283032727788626315832995, −18.49699935114684772092566070324, −17.403052199503284889074528349538, −16.96175391695030884441896280770, −16.10708855971536985130937859540, −15.40732976232400686886743444156, −14.800542473324189793303807951406, −13.48429298345898645588102190981, −12.94918420976006980168922669445, −12.46654169795903811590012495035, −11.50987846924122931231911272427, −10.095052621999026204297972121200, −9.27794582218147069695759851098, −8.32796076438674325416731857328, −7.7232011111032229727953743583, −6.37208172001473919425368128915, −6.06257033752560567583566888695, −4.7654729104772754127241137689, −4.09265631504596282364251117736, −3.22709793846186356549326073317, −1.75528780434596092835529006520, 0.07645289500051716570118760311, 1.75880278193284237088665432923, 2.81768952847267769769172604511, 3.55811013334650982005970433375, 4.24399670704207864028378641954, 5.76816470832638620716490372080, 6.33751878794424762756942355198, 7.110486841409885724863964113441, 8.6764631570075678598358359793, 9.45423402731359758403361801962, 10.3585445994082325742085790070, 11.1075040773067441582775674060, 11.75932141460752281075705170089, 12.66490697811648915715704906082, 13.709567942807355126889999704692, 14.07400203372809207396919466459, 15.03373522501886358215770450168, 15.92703528012409313668914206759, 16.62801033677923717302944777333, 18.174985977238128365966048834078, 18.638038293061409008385692229729, 19.3411301277037197047227709107, 19.93455046259559942831789148097, 21.02369400147650172461480049583, 21.86697409680556502407513109003

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