↑

Properties

Label 1-927-927.481-r1-0-0
Degree $1$
Conductor $927$
Sign $-0.867 - 0.497i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.867 - 0.497i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ -0.867 - 0.497i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005991853663 + 0.02247423609i\)
\(L(\frac12)\) \(\approx\) \(0.005991853663 + 0.02247423609i\)
\(L(1)\) \(\approx\) \(0.6291019099 + 0.08597607231i\)
\(L(1)\) \(\approx\) \(0.6291019099 + 0.08597607231i\)

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 \( 1 + (0.816 + 0.577i)T \)
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

−20.90913638850274355457200565757, −20.37067585875525284510313919656, −19.5486933929864582424844841844, −18.87437028448952814053819321341, −17.90366601994775784043995481927, −17.404814368857843622184751639288, −16.46891561979317028445555788685, −15.78010739360653606749537066501, −15.194175871585729561813709972275, −14.14676972197026165020969210048, −12.942132272470895838387970812394, −12.01312603041741726141459586732, −11.62846345203973090204251885109, −10.58971107049679582678065385206, −9.47287456497512141707076556667, −8.868861246193022532706729232360, −8.336015742143480165521155322417, −7.41366480050777253628240962527, −6.223603010918924704487867078515, −5.55815049527345464871289735221, −4.41948146726420391637034272246, −3.13801359703083535587345360609, −1.97608453776123506641747939286, −1.0912353770780587477774665543, −0.00736291982625635796536562081, 1.42740443954198582170673455115, 2.10846404932929883835174448964, 3.721681980753041701747175011436, 3.95015378018895292168196174146, 5.9800886537710087270170020083, 6.60830265431536393914985701900, 7.33619174503680754263345851455, 8.20676549225205355123665044439, 9.00927021851601238247834598245, 10.25798359841555234722563060007, 10.49531245169175266092815089519, 11.511989675251798422434883018393, 12.01849683499312187803737901400, 13.43796749460034768248812435804, 14.37906471316947879088476412120, 14.94939080721100812249148666947, 15.90688602829556796811970121852, 16.998594318293060685734898318866, 17.17510023760152502285707366446, 18.30865623954808485248526574674, 18.94798452720341758316974208042, 19.57607921032237464305467806704, 20.360499112035057416501955815974, 21.19676641200027488413253055159, 21.997706020566067571101340445614

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