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Properties

Label 1-927-927.628-r1-0-0
Degree $1$
Conductor $927$
Sign $-0.543 - 0.839i$
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  + (0.881 + 0.473i)2-s + (0.552 + 0.833i)4-s + (0.153 − 0.988i)5-s + (−0.908 + 0.417i)7-s + (0.0922 + 0.995i)8-s + (0.602 − 0.798i)10-s + (−0.881 − 0.473i)11-s + (0.816 + 0.577i)13-s + (−0.998 − 0.0615i)14-s + (−0.389 + 0.920i)16-s + (0.739 − 0.673i)17-s + (−0.273 + 0.961i)19-s + (0.908 − 0.417i)20-s + (−0.552 − 0.833i)22-s + (0.881 − 0.473i)23-s + ⋯
L(s)  = 1  + (0.881 + 0.473i)2-s + (0.552 + 0.833i)4-s + (0.153 − 0.988i)5-s + (−0.908 + 0.417i)7-s + (0.0922 + 0.995i)8-s + (0.602 − 0.798i)10-s + (−0.881 − 0.473i)11-s + (0.816 + 0.577i)13-s + (−0.998 − 0.0615i)14-s + (−0.389 + 0.920i)16-s + (0.739 − 0.673i)17-s + (−0.273 + 0.961i)19-s + (0.908 − 0.417i)20-s + (−0.552 − 0.833i)22-s + (0.881 − 0.473i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.543 - 0.839i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (628, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ -0.543 - 0.839i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2936058395 - 0.5401520141i\)
\(L(\frac12)\) \(\approx\) \(0.2936058395 - 0.5401520141i\)
\(L(1)\) \(\approx\) \(1.319702081 + 0.2431261716i\)
\(L(1)\) \(\approx\) \(1.319702081 + 0.2431261716i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.881 + 0.473i)T \)
5 \( 1 + (0.153 - 0.988i)T \)
7 \( 1 + (-0.908 + 0.417i)T \)
11 \( 1 + (-0.881 - 0.473i)T \)
13 \( 1 + (0.816 + 0.577i)T \)
17 \( 1 + (0.739 - 0.673i)T \)
19 \( 1 + (-0.273 + 0.961i)T \)
23 \( 1 + (0.881 - 0.473i)T \)
29 \( 1 + (-0.779 - 0.626i)T \)
31 \( 1 + (-0.992 + 0.122i)T \)
37 \( 1 + (0.982 + 0.183i)T \)
41 \( 1 + (-0.779 + 0.626i)T \)
43 \( 1 + (-0.650 - 0.759i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.273 - 0.961i)T \)
59 \( 1 + (-0.908 - 0.417i)T \)
61 \( 1 + (-0.952 - 0.303i)T \)
67 \( 1 + (0.908 + 0.417i)T \)
71 \( 1 + (-0.932 - 0.361i)T \)
73 \( 1 + (-0.932 - 0.361i)T \)
79 \( 1 + (-0.779 - 0.626i)T \)
83 \( 1 + (-0.908 + 0.417i)T \)
89 \( 1 + (-0.445 - 0.895i)T \)
97 \( 1 + (0.213 - 0.976i)T \)
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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

−21.85345821596316915974144327660, −21.40945454240928758467374457822, −20.30696254675634518376028187047, −19.80230713726097199100009940033, −18.74833423963205266912610034434, −18.39813498178629161345696358567, −17.15366351600399171543253541170, −16.11983521404940557101403527829, −15.23475539385358465081868910024, −14.89037298591442177261706140901, −13.67890093124766521197495277610, −13.16191326555892267592644826551, −12.56167694578981608805170060829, −11.27874036697675993452656566397, −10.66062084425814330797093839466, −10.13781782315698012244091074715, −9.16940220942756028443889175200, −7.563920535055786829129574755150, −6.95728334178240457962459389257, −6.013832616937374719872322302021, −5.33502802541597177019094695238, −4.00244848189863623676480966425, −3.24727938194828896923205497023, −2.61446206161297727519696064994, −1.34787083819882689891370787000, 0.087160932503065896651480015200, 1.64672491367851336232355109456, 2.86820959126688993361331348729, 3.68559026325750619003684461, 4.71968150203596669110962855745, 5.656476359894534950085470388905, 6.085143380104442072997200577772, 7.26200290186027348703999420621, 8.22823193901230504027019793408, 8.92678578002278202083298021474, 9.91091262811202755313574824026, 11.13466028499150473412632102431, 11.995663295904771224504837839153, 12.83462965836584906241199608366, 13.228848891646026966810953280073, 14.05856444304792538478559555780, 15.10689293863790327541825063154, 15.95074944692875516119696393274, 16.49731049195648993963459336692, 16.90019583936577565702260964993, 18.36070637027819975655563519540, 18.91459451241794516637192557683, 20.19511690561657496699186574083, 20.81477408748075445567259631158, 21.36397265942882573702068233630

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