↑

Properties

Label 1-927-927.223-r0-0-0
Degree $1$
Conductor $927$
Sign $-0.770 - 0.637i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.650 − 0.759i)2-s + (−0.153 − 0.988i)4-s + (0.992 + 0.122i)5-s + (−0.850 + 0.526i)7-s + (−0.850 − 0.526i)8-s + (0.739 − 0.673i)10-s + (−0.982 − 0.183i)11-s + (−0.0307 − 0.999i)13-s + (−0.153 + 0.988i)14-s + (−0.952 + 0.303i)16-s + (0.969 + 0.243i)17-s + (0.816 − 0.577i)19-s + (−0.0307 − 0.999i)20-s + (−0.779 + 0.626i)22-s + (0.332 − 0.943i)23-s + ⋯
L(s)  = 1  + (0.650 − 0.759i)2-s + (−0.153 − 0.988i)4-s + (0.992 + 0.122i)5-s + (−0.850 + 0.526i)7-s + (−0.850 − 0.526i)8-s + (0.739 − 0.673i)10-s + (−0.982 − 0.183i)11-s + (−0.0307 − 0.999i)13-s + (−0.153 + 0.988i)14-s + (−0.952 + 0.303i)16-s + (0.969 + 0.243i)17-s + (0.816 − 0.577i)19-s + (−0.0307 − 0.999i)20-s + (−0.779 + 0.626i)22-s + (0.332 − 0.943i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.770 - 0.637i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ -0.770 - 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6090126618 - 1.689867173i\)
\(L(\frac12)\) \(\approx\) \(0.6090126618 - 1.689867173i\)
\(L(1)\) \(\approx\) \(1.142185748 - 0.8172045522i\)
\(L(1)\) \(\approx\) \(1.142185748 - 0.8172045522i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.650 - 0.759i)T \)
5 \( 1 + (0.992 + 0.122i)T \)
7 \( 1 + (-0.850 + 0.526i)T \)
11 \( 1 + (-0.982 - 0.183i)T \)
13 \( 1 + (-0.0307 - 0.999i)T \)
17 \( 1 + (0.969 + 0.243i)T \)
19 \( 1 + (0.816 - 0.577i)T \)
23 \( 1 + (0.332 - 0.943i)T \)
29 \( 1 + (-0.389 - 0.920i)T \)
31 \( 1 + (-0.952 + 0.303i)T \)
37 \( 1 + (0.445 - 0.895i)T \)
41 \( 1 + (-0.602 - 0.798i)T \)
43 \( 1 + (-0.998 - 0.0615i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.908 - 0.417i)T \)
59 \( 1 + (-0.850 - 0.526i)T \)
61 \( 1 + (-0.696 + 0.717i)T \)
67 \( 1 + (-0.0307 + 0.999i)T \)
71 \( 1 + (0.992 - 0.122i)T \)
73 \( 1 + (-0.602 - 0.798i)T \)
79 \( 1 + (0.992 + 0.122i)T \)
83 \( 1 + (-0.0307 - 0.999i)T \)
89 \( 1 + (0.932 - 0.361i)T \)
97 \( 1 + (-0.696 - 0.717i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.07527452568573421284230785540, −21.632942270653588053850876916421, −20.7406388432986198593422301698, −20.14935774311454881423526160336, −18.62412046467601293336032720590, −18.27534639529517097497884629000, −16.97988504564471961176696675793, −16.67915317385651505293203027792, −15.936360315111853568663072155438, −14.93061990768712975537104695237, −13.99237387626905222573141149362, −13.550593138473956744505160109802, −12.81013094121053507401595426161, −12.033862826920606712075452335800, −10.82077201315533228174399620680, −9.68492251850016911375623886157, −9.302959180028928094955529420177, −7.93537222441124750995921426885, −7.19663041919540323550637423911, −6.37716849281498347628350231118, −5.511408195730811656092286232975, −4.87299363698643769202336786623, −3.57962821165031857387108002794, −2.88592549854718531971992318560, −1.539295806899748086219927655223, 0.6044218354667383275193848786, 2.03030186971766427525910463071, 2.85401434080369857963746638395, 3.4262023269216696824105801268, 5.00573393159733200774129296855, 5.59406856372605770319728757991, 6.19813385029312702016573510136, 7.4047413132526706879054627212, 8.77307929238617602101730246630, 9.61571745244509656653688012934, 10.27008520120960230813246823562, 10.878085936533445540385369911558, 12.12194112635753870585032091668, 12.85846881399478932331467617688, 13.28570874745204479561361664041, 14.17947186470541618534947194427, 15.0688406007133093590025444231, 15.7756414634461409111997924824, 16.7681287926279847526776268161, 17.96832011691429233310302387561, 18.48600336299901971846993197298, 19.17583035224880594935923481818, 20.2553373247392388629471309380, 20.79420986159671004837509425592, 21.64698667899760086685583772138

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