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Properties

Label 1-927-927.256-r0-0-0
Degree $1$
Conductor $927$
Sign $-0.820 + 0.571i$
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  + (0.445 + 0.895i)2-s + (−0.602 + 0.798i)4-s + (0.739 + 0.673i)5-s + (0.650 + 0.759i)7-s + (−0.982 − 0.183i)8-s + (−0.273 + 0.961i)10-s + (0.552 − 0.833i)11-s + (0.650 + 0.759i)13-s + (−0.389 + 0.920i)14-s + (−0.273 − 0.961i)16-s + (−0.908 − 0.417i)17-s + (0.881 + 0.473i)19-s + (−0.982 + 0.183i)20-s + (0.992 + 0.122i)22-s + (−0.998 + 0.0615i)23-s + ⋯
L(s)  = 1  + (0.445 + 0.895i)2-s + (−0.602 + 0.798i)4-s + (0.739 + 0.673i)5-s + (0.650 + 0.759i)7-s + (−0.982 − 0.183i)8-s + (−0.273 + 0.961i)10-s + (0.552 − 0.833i)11-s + (0.650 + 0.759i)13-s + (−0.389 + 0.920i)14-s + (−0.273 − 0.961i)16-s + (−0.908 − 0.417i)17-s + (0.881 + 0.473i)19-s + (−0.982 + 0.183i)20-s + (0.992 + 0.122i)22-s + (−0.998 + 0.0615i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.820 + 0.571i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ -0.820 + 0.571i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6464228623 + 2.057939469i\)
\(L(\frac12)\) \(\approx\) \(0.6464228623 + 2.057939469i\)
\(L(1)\) \(\approx\) \(1.086032877 + 1.060246565i\)
\(L(1)\) \(\approx\) \(1.086032877 + 1.060246565i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.445 + 0.895i)T \)
5 \( 1 + (0.739 + 0.673i)T \)
7 \( 1 + (0.650 + 0.759i)T \)
11 \( 1 + (0.552 - 0.833i)T \)
13 \( 1 + (0.650 + 0.759i)T \)
17 \( 1 + (-0.908 - 0.417i)T \)
19 \( 1 + (0.881 + 0.473i)T \)
23 \( 1 + (-0.998 + 0.0615i)T \)
29 \( 1 + (0.739 + 0.673i)T \)
31 \( 1 + (-0.696 + 0.717i)T \)
37 \( 1 + (0.932 - 0.361i)T \)
41 \( 1 + (0.213 + 0.976i)T \)
43 \( 1 + (0.932 + 0.361i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.0307 - 0.999i)T \)
59 \( 1 + (0.650 - 0.759i)T \)
61 \( 1 + (-0.908 - 0.417i)T \)
67 \( 1 + (-0.982 - 0.183i)T \)
71 \( 1 + (0.213 + 0.976i)T \)
73 \( 1 + (0.739 - 0.673i)T \)
79 \( 1 + (-0.952 + 0.303i)T \)
83 \( 1 + (-0.982 + 0.183i)T \)
89 \( 1 + (-0.602 - 0.798i)T \)
97 \( 1 + (0.0922 - 0.995i)T \)
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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

−21.43321494939489233531074170172, −20.57775048012343967347728527321, −20.22055585682133802689982786767, −19.62575534024617658761761719439, −18.16045907433969191592209087698, −17.773860533868354315169798692161, −17.11772183671948935014589715018, −15.86012445921643140744703551435, −14.96291676889855810305510278060, −14.000228808144756957461267994167, −13.52200616999863629453869847358, −12.75268552456125800991600753119, −11.91944777108680112892202616166, −11.01893026374586691139302400685, −10.26009955258107242059303253763, −9.50289875324793213460738144785, −8.69491846416351100183021175208, −7.63964844236933946519103018716, −6.30089430263621348406557550470, −5.53756930585855697065536921281, −4.45875929485890382636975985873, −4.07765968846034615971026672759, −2.60270931459877385654560298747, −1.66917606095070205026083489734, −0.87207329627675369551651218625, 1.576638244476262063476544569561, 2.75180497480692854549491038873, 3.69633229939375080324115671110, 4.79896970771178349838635419129, 5.7678173757150333119132172220, 6.28782427237465649550022556319, 7.13646357395981764495859391111, 8.26069769559627995847532739362, 8.95395225090162810153466767942, 9.70401714006716273652856537668, 11.15251944391246283719591370041, 11.63809839696225436615273048444, 12.74828927593603257164416140498, 13.820703119400384746553058007138, 14.15750130669450683136785327726, 14.82115429275419145576507629275, 15.950616751210015234677554406057, 16.36459344359969117608023693733, 17.53118337811499458755937417295, 18.220045837365442796288483016545, 18.488834977643665453553639208955, 19.821880983918563074784306836047, 21.07080267056661932348477277507, 21.629639921107265386351308457133, 22.09557490049783608709237913708

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