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Properties

Label 1-936-936.259-r1-0-0
Degree $1$
Conductor $936$
Sign $0.173 + 0.984i$
Analytic cond. $100.587$
Root an. cond. $100.587$
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 + (0.5 + 0.866i)11-s + 17-s − 19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + 35-s + 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 + (0.5 + 0.866i)11-s + 17-s − 19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + 35-s + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(100.587\)
Root analytic conductor: \(100.587\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 936,\ (1:\ ),\ 0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.091119494 + 0.9155579654i\)
\(L(\frac12)\) \(\approx\) \(1.091119494 + 0.9155579654i\)
\(L(1)\) \(\approx\) \(0.9292718134 + 0.1489551105i\)
\(L(1)\) \(\approx\) \(0.9292718134 + 0.1489551105i\)

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 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
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 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 - T \)
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

−21.39340355308840720450251981183, −20.83969670738663159972264293409, −19.605462708294075433395981426243, −19.28540213974362480220708913687, −18.523439770319169056937192673517, −17.32858361124229100370360999660, −16.6132821022130747328924916155, −16.02119897108724000981289294214, −15.17006109070689510552786475557, −14.40398509228488545567005658529, −13.1437380968898147611784738911, −12.762762966729569162208981733044, −11.69621787111821105453157320428, −11.298721463720684746466084548379, −9.826468656468049305498368884116, −9.21671521083940152137619452976, −8.36532109097921678249700363976, −7.71512674270038554753893345270, −6.305462063546984154340043182041, −5.74465445424983225525045358784, −4.690538099066209472033163931939, −3.70973066232431268256029492026, −2.8139467031203186509867082443, −1.463147444919176661763356295691, −0.40579724849859025458304049413, 0.83431168218869366503437372381, 2.18704157184462988396294068936, 3.32876251998193746314382575831, 3.97795732724135850923752086276, 4.97336857615239342943831439811, 6.422575541318478816203682696367, 6.90279596404271174641097744258, 7.66232421693726167337956199129, 8.70454249752841326195327488369, 9.867874802198086673903957485068, 10.44690469394532280634529593775, 11.17467491768406566588675837463, 12.3001016020545927853875598411, 12.81590481658856227096559916720, 14.11104240936032424652394524673, 14.5477238280746040935016507370, 15.35973517583392998171811309246, 16.36257276881765023479670150742, 16.99245174924176416774004488046, 17.91710877982355902199235890670, 18.754711595232646619876260875213, 19.50014172987016392481274464776, 20.07428965627780636789944110695, 20.976490605097896201209429457130, 21.98700023900238301654870779875

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