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Properties

Label 1-927-927.340-r1-0-0
Degree $1$
Conductor $927$
Sign $0.224 - 0.974i$
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  + (−0.0307 + 0.999i)2-s + (−0.998 − 0.0615i)4-s + (0.779 − 0.626i)5-s + (0.816 − 0.577i)7-s + (0.0922 − 0.995i)8-s + (0.602 + 0.798i)10-s + (0.0307 − 0.999i)11-s + (−0.908 − 0.417i)13-s + (0.552 + 0.833i)14-s + (0.992 + 0.122i)16-s + (0.739 + 0.673i)17-s + (−0.273 − 0.961i)19-s + (−0.816 + 0.577i)20-s + (0.998 + 0.0615i)22-s + (−0.0307 − 0.999i)23-s + ⋯
L(s)  = 1  + (−0.0307 + 0.999i)2-s + (−0.998 − 0.0615i)4-s + (0.779 − 0.626i)5-s + (0.816 − 0.577i)7-s + (0.0922 − 0.995i)8-s + (0.602 + 0.798i)10-s + (0.0307 − 0.999i)11-s + (−0.908 − 0.417i)13-s + (0.552 + 0.833i)14-s + (0.992 + 0.122i)16-s + (0.739 + 0.673i)17-s + (−0.273 − 0.961i)19-s + (−0.816 + 0.577i)20-s + (0.998 + 0.0615i)22-s + (−0.0307 − 0.999i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.224 - 0.974i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (340, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.224 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.487823301 - 1.184604778i\)
\(L(\frac12)\) \(\approx\) \(1.487823301 - 1.184604778i\)
\(L(1)\) \(\approx\) \(1.138377727 + 0.04722602131i\)
\(L(1)\) \(\approx\) \(1.138377727 + 0.04722602131i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.0307 + 0.999i)T \)
5 \( 1 + (0.779 - 0.626i)T \)
7 \( 1 + (0.816 - 0.577i)T \)
11 \( 1 + (0.0307 - 0.999i)T \)
13 \( 1 + (-0.908 - 0.417i)T \)
17 \( 1 + (0.739 + 0.673i)T \)
19 \( 1 + (-0.273 - 0.961i)T \)
23 \( 1 + (-0.0307 - 0.999i)T \)
29 \( 1 + (-0.153 - 0.988i)T \)
31 \( 1 + (0.389 + 0.920i)T \)
37 \( 1 + (0.982 - 0.183i)T \)
41 \( 1 + (-0.153 + 0.988i)T \)
43 \( 1 + (-0.332 - 0.943i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.273 + 0.961i)T \)
59 \( 1 + (0.816 + 0.577i)T \)
61 \( 1 + (0.213 - 0.976i)T \)
67 \( 1 + (-0.816 - 0.577i)T \)
71 \( 1 + (-0.932 + 0.361i)T \)
73 \( 1 + (-0.932 + 0.361i)T \)
79 \( 1 + (-0.153 - 0.988i)T \)
83 \( 1 + (0.816 - 0.577i)T \)
89 \( 1 + (-0.445 + 0.895i)T \)
97 \( 1 + (-0.952 - 0.303i)T \)
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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

−21.768581997119485584458323118290, −20.993745692443749083137335645294, −20.51810673491157779583049747187, −19.39705372749540819205634002402, −18.66596103382928157798381230578, −18.01033984065494842749982827022, −17.4540524226587808387608721196, −16.58999932416463486434808120367, −14.890464630867252298979094275628, −14.68805637430320769671713871548, −13.824999771022695077311899765791, −12.87469477792548343594804259874, −11.963618150325438190465719938768, −11.50349035497012185814777276658, −10.35254643408109917438226217331, −9.78711737956603861607261937039, −9.13095366384004236844393221594, −7.931774236515671048910431743199, −7.12649100920014795069268924807, −5.685392990966799580336584021833, −5.09353143984261821942728040832, −4.0644896186058831972832354099, −2.83581239957681760143135015925, −2.089908183185779803312539485464, −1.39603588311022184256857536373, 0.42855727776466387486841381014, 1.24176034089432089634726206907, 2.72908906988276351923657049001, 4.18864573016889098891908719481, 4.87618781645009921221173898584, 5.69043344915611972750125290734, 6.46799112312891878687702713597, 7.58845577111400515425896982343, 8.299619994166809840124494192546, 8.985393059098606706134759775630, 10.02064903684063817792658156312, 10.67531133418062771330284599554, 12.01767090554064836927146859436, 12.98292569470720218964304880152, 13.62529128765934514481501839099, 14.38187005822813906893809725827, 14.98710181207382071059657202346, 16.13440470555440523963921743507, 16.89477243748843049488929043304, 17.28622048764289161398770766372, 18.00999741551571433925649312010, 18.98900236709600379036825135259, 19.86995786163823260438218208542, 20.89693987895865309278241409435, 21.64998278824460961014580517046

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