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Properties

Label 1-927-927.2-r1-0-0
Degree $1$
Conductor $927$
Sign $0.967 - 0.254i$
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  + (0.602 + 0.798i)2-s + (−0.273 + 0.961i)4-s + (−0.0922 + 0.995i)5-s + (−0.779 + 0.626i)7-s + (−0.932 + 0.361i)8-s + (−0.850 + 0.526i)10-s + (−0.992 + 0.122i)11-s + (−0.779 + 0.626i)13-s + (−0.969 − 0.243i)14-s + (−0.850 − 0.526i)16-s + (−0.332 − 0.943i)17-s + (−0.998 − 0.0615i)19-s + (−0.932 − 0.361i)20-s + (−0.696 − 0.717i)22-s + (0.389 + 0.920i)23-s + ⋯
L(s)  = 1  + (0.602 + 0.798i)2-s + (−0.273 + 0.961i)4-s + (−0.0922 + 0.995i)5-s + (−0.779 + 0.626i)7-s + (−0.932 + 0.361i)8-s + (−0.850 + 0.526i)10-s + (−0.992 + 0.122i)11-s + (−0.779 + 0.626i)13-s + (−0.969 − 0.243i)14-s + (−0.850 − 0.526i)16-s + (−0.332 − 0.943i)17-s + (−0.998 − 0.0615i)19-s + (−0.932 − 0.361i)20-s + (−0.696 − 0.717i)22-s + (0.389 + 0.920i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.967 - 0.254i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.967 - 0.254i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2520405872 + 0.03259209643i\)
\(L(\frac12)\) \(\approx\) \(-0.2520405872 + 0.03259209643i\)
\(L(1)\) \(\approx\) \(0.5535120165 + 0.7058466442i\)
\(L(1)\) \(\approx\) \(0.5535120165 + 0.7058466442i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.602 + 0.798i)T \)
5 \( 1 + (-0.0922 + 0.995i)T \)
7 \( 1 + (-0.779 + 0.626i)T \)
11 \( 1 + (-0.992 + 0.122i)T \)
13 \( 1 + (-0.779 + 0.626i)T \)
17 \( 1 + (-0.332 - 0.943i)T \)
19 \( 1 + (-0.998 - 0.0615i)T \)
23 \( 1 + (0.389 + 0.920i)T \)
29 \( 1 + (-0.0922 + 0.995i)T \)
31 \( 1 + (0.881 - 0.473i)T \)
37 \( 1 + (0.739 + 0.673i)T \)
41 \( 1 + (-0.816 + 0.577i)T \)
43 \( 1 + (0.739 - 0.673i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.552 + 0.833i)T \)
59 \( 1 + (0.779 + 0.626i)T \)
61 \( 1 + (0.332 + 0.943i)T \)
67 \( 1 + (0.932 - 0.361i)T \)
71 \( 1 + (-0.816 + 0.577i)T \)
73 \( 1 + (0.0922 + 0.995i)T \)
79 \( 1 + (-0.908 + 0.417i)T \)
83 \( 1 + (-0.932 - 0.361i)T \)
89 \( 1 + (0.273 + 0.961i)T \)
97 \( 1 + (-0.982 + 0.183i)T \)
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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

−20.82244777026448330409102128270, −20.28603350940236954107756571493, −19.3561340092058279911895554709, −19.08297284801334516177884991335, −17.66683200470301139548293561293, −17.032745900365896796205336255444, −15.96955278873928869402995696593, −15.33342503082020268214222961114, −14.34989059630236646246100628508, −13.29024508173914941773765447711, −12.79847827768944315557101467667, −12.43997988192198152740453086606, −11.14641757264237864978594471751, −10.348048905793532385646100966600, −9.793436976532088110160730454684, −8.710830579576985675849320360997, −7.86151189067728566050985672242, −6.517455231888995722763524353602, −5.693306176151185578949607413656, −4.69106200829479473007139471523, −4.11003464474602454534461369990, −2.96383681512856892564614869090, −2.063902260670059017943542693844, −0.6819298333685503521411942407, −0.06145245113728490892164628063, 2.47997076725996946076193655125, 2.807309154499169696306022710, 3.98595518071225388203650398142, 5.0299600225142679571709225156, 5.873430035992335517444264428404, 6.847630695226123650994573281061, 7.23443327406649492146453996464, 8.33959395929852019411652703620, 9.32785695036083510851684192918, 10.15613592338672125764206071245, 11.35874621877708873017700515712, 12.06245276956623335974163762994, 13.029781518764519601681574907, 13.64933370062708080837119017358, 14.61764623237132535993979432051, 15.30951852370506401277080846312, 15.78558260722361865848102366927, 16.73574973660487083543991214797, 17.60720760010132469463678921467, 18.525702209845984478824689610, 18.95634369066939136820499348440, 20.09855685520958418990704564976, 21.274615536374173699418522460234, 21.83190694376556047086994341765, 22.42492657107223710248177437028

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