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Properties

Label 1-927-927.142-r1-0-0
Degree $1$
Conductor $927$
Sign $-0.137 + 0.990i$
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  + (−0.952 − 0.303i)2-s + (0.816 + 0.577i)4-s + (−0.881 + 0.473i)5-s + (0.992 + 0.122i)7-s + (−0.602 − 0.798i)8-s + (0.982 − 0.183i)10-s + (0.952 + 0.303i)11-s + (−0.389 − 0.920i)13-s + (−0.908 − 0.417i)14-s + (0.332 + 0.943i)16-s + (0.445 + 0.895i)17-s + (0.932 + 0.361i)19-s + (−0.992 − 0.122i)20-s + (−0.816 − 0.577i)22-s + (−0.952 + 0.303i)23-s + ⋯
L(s)  = 1  + (−0.952 − 0.303i)2-s + (0.816 + 0.577i)4-s + (−0.881 + 0.473i)5-s + (0.992 + 0.122i)7-s + (−0.602 − 0.798i)8-s + (0.982 − 0.183i)10-s + (0.952 + 0.303i)11-s + (−0.389 − 0.920i)13-s + (−0.908 − 0.417i)14-s + (0.332 + 0.943i)16-s + (0.445 + 0.895i)17-s + (0.932 + 0.361i)19-s + (−0.992 − 0.122i)20-s + (−0.816 − 0.577i)22-s + (−0.952 + 0.303i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.137 + 0.990i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ -0.137 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6901267136 + 0.7924993594i\)
\(L(\frac12)\) \(\approx\) \(0.6901267136 + 0.7924993594i\)
\(L(1)\) \(\approx\) \(0.7102472800 + 0.09587399983i\)
\(L(1)\) \(\approx\) \(0.7102472800 + 0.09587399983i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.952 - 0.303i)T \)
5 \( 1 + (-0.881 + 0.473i)T \)
7 \( 1 + (0.992 + 0.122i)T \)
11 \( 1 + (0.952 + 0.303i)T \)
13 \( 1 + (-0.389 - 0.920i)T \)
17 \( 1 + (0.445 + 0.895i)T \)
19 \( 1 + (0.932 + 0.361i)T \)
23 \( 1 + (-0.952 + 0.303i)T \)
29 \( 1 + (-0.0307 + 0.999i)T \)
31 \( 1 + (-0.650 + 0.759i)T \)
37 \( 1 + (0.273 + 0.961i)T \)
41 \( 1 + (-0.0307 - 0.999i)T \)
43 \( 1 + (-0.969 + 0.243i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.932 - 0.361i)T \)
59 \( 1 + (0.992 - 0.122i)T \)
61 \( 1 + (0.552 - 0.833i)T \)
67 \( 1 + (-0.992 + 0.122i)T \)
71 \( 1 + (0.850 - 0.526i)T \)
73 \( 1 + (0.850 - 0.526i)T \)
79 \( 1 + (-0.0307 + 0.999i)T \)
83 \( 1 + (0.992 + 0.122i)T \)
89 \( 1 + (-0.0922 - 0.995i)T \)
97 \( 1 + (-0.998 + 0.0615i)T \)
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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

−21.1936977953212685758702447606, −20.33952574010361021268838677471, −19.89704305135729949225988573689, −19.010223936471115474076997571161, −18.3244025680532738777967201345, −17.461427064044820105932898920543, −16.561105103863499872325111316, −16.26313698547601847475000870713, −15.15082648235660023581658458727, −14.48433721980921618983030570781, −13.71805507465108434468346261, −12.01357661129420289938270647724, −11.69823978586318052937612699483, −11.10655653968395052688369945280, −9.768168432454075998496617783259, −9.14547067592231469213649453816, −8.2671791262306439898582828905, −7.57679262775232254295979288164, −6.89539854887934742943066443055, −5.67196842613926346165728512013, −4.715498430142074698821747085600, −3.77158367433549353828319814653, −2.30318035274964435601110784574, −1.24578712544382854155910775841, −0.378519018094842371980523482362, 1.03803635299249104245323996970, 1.88860585174535802949179828137, 3.21274860327010849609176889743, 3.852918456367964868875034642396, 5.16994437323930046057340331739, 6.40389309026686732991347793209, 7.39919609500454274260946206698, 7.946776884776775843397150375747, 8.64081102437194511066300015563, 9.76955432717370953666617492676, 10.55586162240771092391217728337, 11.30006150221717110323283093665, 12.06220995586181963746492194036, 12.53113686877225713515414434580, 14.19527980355778263789293077019, 14.8374590803198035339187445645, 15.56523130617039739075300340908, 16.45459361352062975427460099020, 17.35629280200361934567518072684, 17.984388802892377860374620000493, 18.653737482105236104221823981531, 19.63094885526216587571016668779, 20.06305651537684152842004460029, 20.794209275389484223600880307939, 21.98391830503340174393727169486

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