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Properties

Label 1-837-837.209-r0-0-0
Degree $1$
Conductor $837$
Sign $0.949 - 0.314i$
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 + (0.939 − 0.342i)5-s + (−0.882 + 0.469i)7-s + (0.978 − 0.207i)8-s + (0.913 − 0.406i)10-s + (0.0348 − 0.999i)11-s + (0.997 + 0.0697i)13-s + (−0.848 + 0.529i)14-s + (0.961 − 0.275i)16-s + (0.669 + 0.743i)17-s + (−0.104 + 0.994i)19-s + (0.882 − 0.469i)20-s + (−0.0348 − 0.999i)22-s + (−0.882 − 0.469i)23-s + ⋯
L(s)  = 1  + (0.997 − 0.0697i)2-s + (0.990 − 0.139i)4-s + (0.939 − 0.342i)5-s + (−0.882 + 0.469i)7-s + (0.978 − 0.207i)8-s + (0.913 − 0.406i)10-s + (0.0348 − 0.999i)11-s + (0.997 + 0.0697i)13-s + (−0.848 + 0.529i)14-s + (0.961 − 0.275i)16-s + (0.669 + 0.743i)17-s + (−0.104 + 0.994i)19-s + (0.882 − 0.469i)20-s + (−0.0348 − 0.999i)22-s + (−0.882 − 0.469i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.949 - 0.314i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.949 - 0.314i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.254915807 - 0.5255627773i\)
\(L(\frac12)\) \(\approx\) \(3.254915807 - 0.5255627773i\)
\(L(1)\) \(\approx\) \(2.188180270 - 0.2130045192i\)
\(L(1)\) \(\approx\) \(2.188180270 - 0.2130045192i\)

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 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (-0.882 + 0.469i)T \)
11 \( 1 + (0.0348 - 0.999i)T \)
13 \( 1 + (0.997 + 0.0697i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.882 - 0.469i)T \)
29 \( 1 + (-0.997 + 0.0697i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.241 + 0.970i)T \)
43 \( 1 + (-0.559 - 0.829i)T \)
47 \( 1 + (0.719 + 0.694i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.559 + 0.829i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (0.374 + 0.927i)T \)
83 \( 1 + (-0.997 + 0.0697i)T \)
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

−22.19224651955730165562200162857, −21.61567882869426801315890450911, −20.48579585095425668069135021302, −20.28149421368536705037749242870, −19.08162648701253977302265476205, −18.13877716269071459713148485525, −17.226792034007748979953002863627, −16.44807320860829909862599304626, −15.61756143753943605483638510925, −14.856916405694816272646824636837, −13.81010448898384714323663532446, −13.45055665404885983865653645611, −12.68137406519932863004693985931, −11.70594124388368740046162224255, −10.72062599300496559378039875481, −9.98220901458974320380701409988, −9.20285064836050788102856416649, −7.62068420597931437004096241317, −6.908374868266195752968026037942, −6.15599728232556893191948528430, −5.37939573193308480826320431161, −4.27922110418294639348163566623, −3.33066357276261369008954060828, −2.4755824621544474995886492433, −1.39114070589808314823963285108, 1.27012449779415783493445986471, 2.26752556238993435048376629398, 3.3523351642967928228349228255, 4.03207746250745898245695593509, 5.56402295952677560856710764957, 5.90096327921237718484045020411, 6.501743935676831276716036400416, 7.95575343635737537677461543656, 8.880437181802706746635628435030, 9.95356495472615545750530860858, 10.63712816963811173112816836249, 11.67458499947747285415139559258, 12.66086698100290926263857004798, 13.07755557573101368526271714453, 13.96654351799920478724835197452, 14.56849221798409397577201266042, 15.719371274694469874136710981875, 16.44326761099014765282646614578, 16.852257232388778351056793144221, 18.34523144098994176903675163403, 18.920322422146035746495222886061, 19.923631591520080429571828546433, 20.79234764492662243660026075446, 21.43038353759609594631807266523, 21.99562889522363262359699620421

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