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Properties

Label 1-927-927.365-r1-0-0
Degree $1$
Conductor $927$
Sign $-0.893 - 0.449i$
Analytic cond. $99.6199$
Root an. cond. $99.6199$
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  − 2-s + 4-s − 5-s + (−0.5 − 0.866i)7-s − 8-s + 10-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + 16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s − 20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + ⋯
L(s)  = 1  − 2-s + 4-s − 5-s + (−0.5 − 0.866i)7-s − 8-s + 10-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + 16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s − 20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.893 - 0.449i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ -0.893 - 0.449i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1280697068 - 0.5391553477i\)
\(L(\frac12)\) \(\approx\) \(0.1280697068 - 0.5391553477i\)
\(L(1)\) \(\approx\) \(0.5293854368 - 0.1336491209i\)
\(L(1)\) \(\approx\) \(0.5293854368 - 0.1336491209i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
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 \( 1 + T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 + T \)
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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

−21.731133884652179891154224478506, −21.262696169853702018831204535937, −20.05145227836405262076663682133, −19.4644455941023824947284679593, −18.72563446833504098176730882762, −18.55218895132341585457231308106, −16.94560320363233067185334843883, −16.61277227867284775000780206435, −15.89123484323600396806419249069, −14.89652294608259277944299839878, −14.48556785672875168199100531365, −12.71009283101231778062410581023, −12.2352436134737669090587501306, −11.39616491406615679482741357102, −10.722871992941999390728893524781, −9.62109586023780117255572238506, −8.80827847578085032885714822801, −8.29395114737773330307042922121, −7.294210475242619936158767550807, −6.41754380464714433190966145732, −5.65842114692856304413798079342, −4.103339520395240900985100406257, −3.23890704690565033379056556078, −2.20909441132964482951106133031, −0.97618885647739050379396442995, 0.24640484518913980745307094646, 0.92278958456073478691497587177, 2.420851348986634335192540331123, 3.401868836457976709965346925371, 4.32348075698985322426432752349, 5.59518358285877778905388735528, 6.98801531671337560640730320083, 7.30163641798701567855933140334, 7.99688534453092466029269330248, 9.30012444676957808489023337726, 9.70414708344745287538084943738, 10.846966833440182578072838595913, 11.346789143737423085820254615979, 12.38142994861256084106484978484, 13.008123144975265208461504407250, 14.443411373984041095732823017000, 15.22762859697000243914276220343, 15.81093799401739709026706628074, 16.78305451501052115551944651027, 17.26054504019597621633666873652, 18.16719605249326099637921590652, 19.146024751336239935122109781690, 19.70259195569977908377604365388, 20.24193718705046636697932625050, 20.89534357406769467358423411235

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