Properties

Label 1-927-927.169-r0-0-0
Degree $1$
Conductor $927$
Sign $-0.821 - 0.570i$
Analytic cond. $4.30496$
Root an. cond. $4.30496$
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  + (0.213 − 0.976i)2-s + (−0.908 − 0.417i)4-s + (−0.0307 − 0.999i)5-s + (−0.389 + 0.920i)7-s + (−0.602 + 0.798i)8-s + (−0.982 − 0.183i)10-s + (0.213 − 0.976i)11-s + (0.992 − 0.122i)13-s + (0.816 + 0.577i)14-s + (0.650 + 0.759i)16-s + (0.445 − 0.895i)17-s + (0.932 − 0.361i)19-s + (−0.389 + 0.920i)20-s + (−0.908 − 0.417i)22-s + (0.213 + 0.976i)23-s + ⋯
L(s)  = 1  + (0.213 − 0.976i)2-s + (−0.908 − 0.417i)4-s + (−0.0307 − 0.999i)5-s + (−0.389 + 0.920i)7-s + (−0.602 + 0.798i)8-s + (−0.982 − 0.183i)10-s + (0.213 − 0.976i)11-s + (0.992 − 0.122i)13-s + (0.816 + 0.577i)14-s + (0.650 + 0.759i)16-s + (0.445 − 0.895i)17-s + (0.932 − 0.361i)19-s + (−0.389 + 0.920i)20-s + (−0.908 − 0.417i)22-s + (0.213 + 0.976i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.821 - 0.570i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ -0.821 - 0.570i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4234338108 - 1.352969517i\)
\(L(\frac12)\) \(\approx\) \(0.4234338108 - 1.352969517i\)
\(L(1)\) \(\approx\) \(0.8169299416 - 0.7335683292i\)
\(L(1)\) \(\approx\) \(0.8169299416 - 0.7335683292i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.213 - 0.976i)T \)
5 \( 1 + (-0.0307 - 0.999i)T \)
7 \( 1 + (-0.389 + 0.920i)T \)
11 \( 1 + (0.213 - 0.976i)T \)
13 \( 1 + (0.992 - 0.122i)T \)
17 \( 1 + (0.445 - 0.895i)T \)
19 \( 1 + (0.932 - 0.361i)T \)
23 \( 1 + (0.213 + 0.976i)T \)
29 \( 1 + (0.881 + 0.473i)T \)
31 \( 1 + (0.332 - 0.943i)T \)
37 \( 1 + (-0.273 + 0.961i)T \)
41 \( 1 + (0.881 - 0.473i)T \)
43 \( 1 + (-0.696 + 0.717i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.932 - 0.361i)T \)
59 \( 1 + (-0.389 - 0.920i)T \)
61 \( 1 + (-0.998 + 0.0615i)T \)
67 \( 1 + (-0.389 - 0.920i)T \)
71 \( 1 + (-0.850 - 0.526i)T \)
73 \( 1 + (-0.850 - 0.526i)T \)
79 \( 1 + (0.881 + 0.473i)T \)
83 \( 1 + (-0.389 + 0.920i)T \)
89 \( 1 + (0.0922 - 0.995i)T \)
97 \( 1 + (0.552 - 0.833i)T \)
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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

−22.551293339276712359728034895474, −21.58569936921913278027806078114, −20.74419916501779550379774596593, −19.665603729804938705581260684680, −18.8717715923832113649744868385, −17.97584628767880106957978822695, −17.50631531565870592183296655259, −16.47505489723437324906078054863, −15.86657323554760322806693107148, −14.938831075200244678733936432289, −14.28199000643560380016416232185, −13.673393615745689422663000052312, −12.75348691337745400479032217641, −11.8712253430974950831725090538, −10.518345666635041111962766095876, −10.11760105259966471360367788510, −9.0120687446702130436945685157, −7.93025105916832783518119365857, −7.22746474410085622022573310578, −6.5351693946348782008216943317, −5.85767554220463844134166886378, −4.48350357337956956135486473131, −3.80554731469418122986926477503, −2.93899631814504057137137849616, −1.2331007472915699555276719194, 0.69362333657227313612166411058, 1.578736014706533276784130563563, 2.95479022388864998180681933, 3.50999980114408082863133166594, 4.79843347332299745708974750663, 5.48872814663653748831686477142, 6.21449057995311524986131521521, 7.91805231542792660664881704405, 8.7842757972828938271582942571, 9.25543382967509492191731369343, 10.092043294984864751556057650623, 11.50090463570436971103414585183, 11.649596669973752572143702652102, 12.63999009202635877138121335555, 13.506517730750658998720357717283, 13.85658171998821375465690812294, 15.245085321385004481182173852227, 15.97548698523185499530197108182, 16.74142871974944137953201708480, 17.93288944489133650653414264568, 18.47978383733531828219411394419, 19.36244463712390879156994612402, 19.95924626208793207898779100950, 20.93016084875219287916001431197, 21.321224235692451833160640884793

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