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Properties

Label 1-837-837.677-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.677 + 0.735i$
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  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.939 − 0.342i)7-s + (0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + (−0.939 − 0.342i)11-s + (−0.173 + 0.984i)13-s + (−0.766 − 0.642i)14-s + (0.173 + 0.984i)16-s + 17-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)20-s + (−0.766 − 0.642i)22-s + (0.766 + 0.642i)23-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.939 − 0.342i)7-s + (0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + (−0.939 − 0.342i)11-s + (−0.173 + 0.984i)13-s + (−0.766 − 0.642i)14-s + (0.173 + 0.984i)16-s + 17-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)20-s + (−0.766 − 0.642i)22-s + (0.766 + 0.642i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.677 + 0.735i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.677 + 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4671688291 + 1.065052933i\)
\(L(\frac12)\) \(\approx\) \(0.4671688291 + 1.065052933i\)
\(L(1)\) \(\approx\) \(1.152146358 + 0.3956203368i\)
\(L(1)\) \(\approx\) \(1.152146358 + 0.3956203368i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (-0.939 - 0.342i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (0.766 - 0.642i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.173 + 0.984i)T \)
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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

−22.15560590494864111037356588177, −21.05363328454960973956098690116, −20.35732026564070716377088422619, −19.45709099570762369007114557017, −18.94398255854097673602030641585, −18.16348005864011095737241828461, −16.766772319638598128340031925936, −15.86000635704314055469173783963, −15.24263375482636623440910775189, −14.79278212594881644490869343592, −13.59291216501371615287310844818, −12.7352239708183331415878095093, −12.35015013122157741718438049039, −11.25043743528180846696121235461, −10.4714311667554933209844951634, −9.902916307107074708526110667571, −8.47486582364760294210616678084, −7.315126146104561617008601604086, −6.78959042620329899940393916770, −5.623465047557972136573211268733, −4.92616533902007771305803433917, −3.599364606343019968926195333034, −3.081296219055797191612441606659, −2.239421429528964527682137683605, −0.35871061684586606810372326470, 1.55670308974480335709254537460, 3.05263692630795183389905452299, 3.66702489064770220450938144226, 4.61617733422255775145812804333, 5.49043206043324785400243748739, 6.41948909415395102155291447789, 7.433680483984770109132396569182, 7.9993096602325920514157455048, 9.11952043245832101367977303227, 10.22915448641833980482436234081, 11.282318309996528997430164373680, 12.03671417665631924212586296298, 12.86313191200640731804298660964, 13.331710155152926500862259819116, 14.371748584608386501873459691429, 15.22709301253216279593494371616, 16.083364141736949997853009262192, 16.52904908262878518603219437626, 17.13826523727850415029726522292, 18.79404975634101974389888863754, 19.264803871945094497708713737138, 20.244247771025222095232196059670, 21.01343017895925491416982690814, 21.53264482309272102717598271091, 22.77039616027086076919604165588

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