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Properties

Label 1-927-927.140-r0-0-0
Degree $1$
Conductor $927$
Sign $0.990 - 0.137i$
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  + (0.952 − 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

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

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.990 - 0.137i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ 0.990 - 0.137i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.489197698 - 0.2409227831i\)
\(L(\frac12)\) \(\approx\) \(3.489197698 - 0.2409227831i\)
\(L(1)\) \(\approx\) \(2.264658954 - 0.2070568910i\)
\(L(1)\) \(\approx\) \(2.264658954 - 0.2070568910i\)

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

−21.76370818899543474220265495166, −21.21445053746723815942199461108, −20.50242090282629408619529356663, −20.045865911893838713746174751418, −18.39270852505203599082268563509, −17.89049754622714263351506918234, −16.99337529673705415022535038431, −16.28206966794869002969240731969, −15.33626742255284414909906357078, −14.69461413131277967803271007310, −13.68044232136911971848864500353, −13.3342467671669254711926683504, −12.39613072938644313894940809176, −11.53889957486052238241166017112, −10.7003365226778424407418977059, −9.77419925297637484297256299577, −8.5062581638079904124088905143, −7.870616158842960815811280464035, −6.91507247760697219879977326509, −5.73206985066169641081723936008, −5.1454551785280482866235775731, −4.67464015143376653700787120978, −3.09433093734768716548712537385, −2.426430080928691059566113278889, −1.25590314625841367278292930584, 1.51516463842736591531485651149, 2.139739597637225181892790889862, 3.09090591656560536339544213439, 4.30695891986975896053430869663, 5.15019060188181469410164653675, 5.74710933612424346980775505236, 6.9688580316450516482607419890, 7.45441149824346694046866509178, 8.92027582249306495632761539154, 9.885684951875077846740508839864, 10.8036788380120433456638745168, 11.20338397058700176571503008981, 12.288238347283028575097289482739, 13.214105353137947085859066664070, 13.76363655318751981884723763742, 14.679308265123994249111447073407, 15.03035193526131174322654080073, 16.169120066334431607266633896762, 17.12398921615889593968622533656, 17.97145446855860815055352916927, 18.680737513616311338732587124998, 19.67627439575532163748491769995, 20.58393162446626565220497780746, 21.22500198836436590498471493264, 21.71983171711480067308141946571

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