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Properties

Label 1-936-936.205-r0-0-0
Degree $1$
Conductor $936$
Sign $0.329 - 0.944i$
Analytic cond. $4.34676$
Root an. cond. $4.34676$
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  + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + 11-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)35-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s + (0.5 + 0.866i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + 11-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)35-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s + (0.5 + 0.866i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.329 - 0.944i$
Analytic conductor: \(4.34676\)
Root analytic conductor: \(4.34676\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 936,\ (0:\ ),\ 0.329 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9586895726 - 0.6804838659i\)
\(L(\frac12)\) \(\approx\) \(0.9586895726 - 0.6804838659i\)
\(L(1)\) \(\approx\) \(0.9840240685 - 0.1326505559i\)
\(L(1)\) \(\approx\) \(0.9840240685 - 0.1326505559i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−21.9082048069526423152430570311, −21.17639400642574263779978991369, −20.48377658626368213783802145109, −19.4259967565314707526172469174, −19.18384756571122162345080283148, −17.8726125967906649851831683745, −17.30394281095075391668360993496, −16.423188957893874076908359807129, −15.64018723173876008230522145208, −14.88809847670898960125999395541, −14.1409848378359137167173386665, −12.96943927338232418547066147138, −12.284012159794451464605016303822, −11.69172747442060536557654082476, −10.831388018144333435072290606774, −9.58653208770551775488292438316, −8.80228837594001023350251915170, −8.28627532982703156164633230688, −7.28744745663521606857193003542, −6.05162215299169998106033331650, −5.39952464772786654197580869796, −4.26071476438883515381308132039, −3.67162986498405377892155324021, −2.08405629228480752172049929458, −1.33321812853195769106621524704, 0.5356630595962446302615353991, 1.96830343318021045209680901445, 3.03880664934381144172788781456, 4.11254467771635810833736688190, 4.62225686080177855821691066915, 6.159820467235006524543551943223, 6.89524807958075488898422329930, 7.54856001238014227720491649792, 8.51793049631417258243159975616, 9.53680330831009755197508652980, 10.47129936770816739767774029087, 11.291437807728580890819103447557, 11.68191323336814062889043403602, 12.93424709134924938573136320910, 13.91238196719559594308766113508, 14.439007856004004854958637085516, 15.22797252818139725306221486728, 16.094414975083675705200467357888, 17.08960146468616042379892238606, 17.64410350576540743617752162204, 18.62269051858770945914898513759, 19.29038356121585071443341667503, 20.183899036464702512266205406962, 20.65330081987383227336683616466, 21.97492321370378893979976545531

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