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Properties

Label 1-927-927.821-r1-0-0
Degree $1$
Conductor $927$
Sign $0.864 - 0.503i$
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  + (0.998 − 0.0615i)2-s + (0.992 − 0.122i)4-s + (−0.213 − 0.976i)5-s + (0.332 + 0.943i)7-s + (0.982 − 0.183i)8-s + (−0.273 − 0.961i)10-s + (0.998 − 0.0615i)11-s + (0.650 − 0.759i)13-s + (0.389 + 0.920i)14-s + (0.969 − 0.243i)16-s + (−0.0922 + 0.995i)17-s + (−0.850 − 0.526i)19-s + (−0.332 − 0.943i)20-s + (0.992 − 0.122i)22-s + (0.998 + 0.0615i)23-s + ⋯
L(s)  = 1  + (0.998 − 0.0615i)2-s + (0.992 − 0.122i)4-s + (−0.213 − 0.976i)5-s + (0.332 + 0.943i)7-s + (0.982 − 0.183i)8-s + (−0.273 − 0.961i)10-s + (0.998 − 0.0615i)11-s + (0.650 − 0.759i)13-s + (0.389 + 0.920i)14-s + (0.969 − 0.243i)16-s + (−0.0922 + 0.995i)17-s + (−0.850 − 0.526i)19-s + (−0.332 − 0.943i)20-s + (0.992 − 0.122i)22-s + (0.998 + 0.0615i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.864 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.864 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.864 - 0.503i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (821, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.864 - 0.503i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.149705806 - 1.390027976i\)
\(L(\frac12)\) \(\approx\) \(5.149705806 - 1.390027976i\)
\(L(1)\) \(\approx\) \(2.310897156 - 0.3435129412i\)
\(L(1)\) \(\approx\) \(2.310897156 - 0.3435129412i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.998 - 0.0615i)T \)
5 \( 1 + (-0.213 - 0.976i)T \)
7 \( 1 + (0.332 + 0.943i)T \)
11 \( 1 + (0.998 - 0.0615i)T \)
13 \( 1 + (0.650 - 0.759i)T \)
17 \( 1 + (-0.0922 + 0.995i)T \)
19 \( 1 + (-0.850 - 0.526i)T \)
23 \( 1 + (0.998 + 0.0615i)T \)
29 \( 1 + (0.952 + 0.303i)T \)
31 \( 1 + (-0.696 - 0.717i)T \)
37 \( 1 + (0.932 + 0.361i)T \)
41 \( 1 + (0.952 - 0.303i)T \)
43 \( 1 + (-0.779 - 0.626i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.850 + 0.526i)T \)
59 \( 1 + (-0.332 + 0.943i)T \)
61 \( 1 + (-0.908 + 0.417i)T \)
67 \( 1 + (0.332 - 0.943i)T \)
71 \( 1 + (-0.739 - 0.673i)T \)
73 \( 1 + (0.739 + 0.673i)T \)
79 \( 1 + (-0.952 - 0.303i)T \)
83 \( 1 + (-0.332 - 0.943i)T \)
89 \( 1 + (0.602 - 0.798i)T \)
97 \( 1 + (0.816 - 0.577i)T \)
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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

−21.65880153055664084389029140841, −21.23272209851990141721523869207, −20.183455327410909663520610653421, −19.58910613866176569181099740060, −18.7239136409735915663556514690, −17.689828373885909321800030293452, −16.73710075282947867864003983619, −16.15698558245461752439331468116, −15.057493673959659359424907678123, −14.38847184569281858333010054974, −13.968498363090572977592082110710, −13.09508309718499811223031998219, −11.96212772325319285531022452042, −11.25161708353535222413214239286, −10.78034003811457761750791165786, −9.736708561433506681963567599725, −8.42172754560822643612034110197, −7.291698948350508850198091245575, −6.80163716391131294271319841408, −6.11101866391481869526281832868, −4.725692677004880416412752083181, −4.025914499408847692219129143123, −3.30446002675704722439563059835, −2.171502178994874899441208573301, −1.05887334030983560582878072012, 0.94865329670718988272121172483, 1.83858267535930546389972443543, 2.991840125179486976394794295043, 4.07230172197301246588139062245, 4.73458505061702486210238427746, 5.76634787892839339029060876109, 6.256272918038907760341368219850, 7.56532578805269830875908281695, 8.55897316900213375990967800443, 9.11660375497938115883545721969, 10.54594557941598860546032109253, 11.34010160013361349080608347391, 12.10767556820881108907033149133, 12.79692567493002770859950615224, 13.34279756715016006712046846680, 14.54145134853587059375809763575, 15.189619542457390493477171402410, 15.749947437345960909069118603389, 16.80848438428627298268880242432, 17.33483927165306647843839513604, 18.61279278676359299325084788271, 19.62584420304279560852835290050, 20.02068556965858527855524977955, 21.11899134665327257112831999101, 21.47150789741087650952264116895

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