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Properties

Label 1-927-927.436-r1-0-0
Degree $1$
Conductor $927$
Sign $-0.325 - 0.945i$
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  + (0.816 + 0.577i)2-s + (0.332 + 0.943i)4-s + (−0.552 + 0.833i)5-s + (0.969 + 0.243i)7-s + (−0.273 + 0.961i)8-s + (−0.932 + 0.361i)10-s + (−0.816 − 0.577i)11-s + (−0.696 + 0.717i)13-s + (0.650 + 0.759i)14-s + (−0.779 + 0.626i)16-s + (−0.602 + 0.798i)17-s + (0.739 + 0.673i)19-s + (−0.969 − 0.243i)20-s + (−0.332 − 0.943i)22-s + (0.816 − 0.577i)23-s + ⋯
L(s)  = 1  + (0.816 + 0.577i)2-s + (0.332 + 0.943i)4-s + (−0.552 + 0.833i)5-s + (0.969 + 0.243i)7-s + (−0.273 + 0.961i)8-s + (−0.932 + 0.361i)10-s + (−0.816 − 0.577i)11-s + (−0.696 + 0.717i)13-s + (0.650 + 0.759i)14-s + (−0.779 + 0.626i)16-s + (−0.602 + 0.798i)17-s + (0.739 + 0.673i)19-s + (−0.969 − 0.243i)20-s + (−0.332 − 0.943i)22-s + (0.816 − 0.577i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.325 - 0.945i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (436, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ -0.325 - 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.7125923224 + 0.9985456965i\)
\(L(\frac12)\) \(\approx\) \(-0.7125923224 + 0.9985456965i\)
\(L(1)\) \(\approx\) \(0.9868331302 + 0.8775806889i\)
\(L(1)\) \(\approx\) \(0.9868331302 + 0.8775806889i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.816 + 0.577i)T \)
5 \( 1 + (-0.552 + 0.833i)T \)
7 \( 1 + (0.969 + 0.243i)T \)
11 \( 1 + (-0.816 - 0.577i)T \)
13 \( 1 + (-0.696 + 0.717i)T \)
17 \( 1 + (-0.602 + 0.798i)T \)
19 \( 1 + (0.739 + 0.673i)T \)
23 \( 1 + (0.816 - 0.577i)T \)
29 \( 1 + (-0.998 - 0.0615i)T \)
31 \( 1 + (0.153 + 0.988i)T \)
37 \( 1 + (0.850 - 0.526i)T \)
41 \( 1 + (-0.998 + 0.0615i)T \)
43 \( 1 + (-0.881 + 0.473i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.739 - 0.673i)T \)
59 \( 1 + (0.969 - 0.243i)T \)
61 \( 1 + (-0.389 - 0.920i)T \)
67 \( 1 + (-0.969 + 0.243i)T \)
71 \( 1 + (-0.445 + 0.895i)T \)
73 \( 1 + (-0.445 + 0.895i)T \)
79 \( 1 + (-0.998 - 0.0615i)T \)
83 \( 1 + (0.969 + 0.243i)T \)
89 \( 1 + (0.982 - 0.183i)T \)
97 \( 1 + (0.992 - 0.122i)T \)
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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

−20.84592584150641971546650809582, −20.42036324059442278275846882212, −20.00440045896465563778742485284, −18.92114464279322716889048729409, −18.02331565112957470036976650991, −17.20244210226004334266064127290, −16.13169260313453139152993256612, −15.114283045032264139545350191791, −15.03883178885080158963025968432, −13.449612073304981544875255265298, −13.297028903095308375229517134040, −12.16630738276214572945431918054, −11.56088877964664306565251133914, −10.85448279557398864285763227593, −9.81124075749947208250770881905, −8.99311964435563690954772632172, −7.661454312129689457947695113616, −7.28393907738643499255335234665, −5.636146809167720753718621519476, −4.86119904282944555553592970311, −4.582069340828532927812004638170, −3.262707180175049727085933080653, −2.28589295690017824001560090479, −1.18017312794113348427735987574, −0.18834520789734837543126119112, 1.89280144041529551275446745548, 2.82815544991778117100284700543, 3.755236050155429647938168475584, 4.711851663205961201948835766769, 5.48061832789130722951470847025, 6.52288363509152058019932617407, 7.33195144907085862942760101038, 8.05058206145357989724683754329, 8.76897234452178347137022683274, 10.30927871212372142494457805920, 11.220162752112168079389070388247, 11.67676143025533909031058115594, 12.65358011342913267182726400600, 13.58465785730191313734831283610, 14.54631804943259708582760542316, 14.79305113987496834227013145583, 15.69904508538278300006732447413, 16.48224317024684982215151171836, 17.3671848469913891133555185281, 18.26860221105231777002120116936, 18.892043344189453459437653043904, 20.0250270350436112611701993765, 20.88165146275953412077713031031, 21.720302946950485895550034816413, 22.074653559465440857763949369560

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