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Properties

Label 1-684-684.331-r0-0-0
Degree $1$
Conductor $684$
Sign $0.564 + 0.825i$
Analytic cond. $3.17648$
Root an. cond. $3.17648$
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  + 5-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)23-s + 25-s − 29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)35-s − 37-s − 41-s + (0.5 − 0.866i)43-s − 47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + 5-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)23-s + 25-s − 29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)35-s − 37-s − 41-s + (0.5 − 0.866i)43-s − 47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.564 + 0.825i$
Analytic conductor: \(3.17648\)
Root analytic conductor: \(3.17648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 684,\ (0:\ ),\ 0.564 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.621835457 + 0.8552563043i\)
\(L(\frac12)\) \(\approx\) \(1.621835457 + 0.8552563043i\)
\(L(1)\) \(\approx\) \(1.316574921 + 0.2953979844i\)
\(L(1)\) \(\approx\) \(1.316574921 + 0.2953979844i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−22.42996889675455202659479487548, −21.84363232308277352286786246051, −20.79658000510949083831181053049, −20.42854804151496661554687232478, −19.313772939319677268991089861022, −18.41216041764479300134847741941, −17.52207838983037722715760081041, −17.01265451327662323780593982615, −16.21369699818455690965064522917, −14.89731026514661447244551170757, −14.34467707034959554404975549784, −13.264574362641262115428926234211, −13.03278568264518850327553159082, −11.48412117416224894375607507455, −10.734997864262851535983861043080, −10.12460275809997398406159383156, −8.92382272778573673297936684244, −8.26554767088315383857759154382, −7.06838092534365188013025446998, −6.154331053468026112255027724277, −5.39077149724216959066146716613, −4.20409243169899260424998612373, −3.23696753085650826225905991122, −1.92241260773460846386044073088, −0.93914162790749542983839378709, 1.630779939923485690356361454442, 2.095926869108476477410258997367, 3.46083161348504942356871700764, 4.82101361603662143818400461284, 5.4311534451940835602211282534, 6.55097143021968735236902231665, 7.257428296061487623809670048244, 8.79157807863114289288365417893, 9.148640852140442887399266227642, 10.06815948593578282508825034178, 11.242621490071469212078407328471, 11.895476525920992417400760940090, 12.91749931767462311695641520768, 13.77364310917541245773093821786, 14.53209501044362562476330170176, 15.32292688089062185927339769672, 16.30583629941337456754553693608, 17.277682509528266558080122509910, 17.93632303293872271600066559362, 18.55977458870018007787971090198, 19.549703005090540188785357451122, 20.71827499999581491630995643469, 21.09166741967310648894126644818, 22.069020138963729762849878443781, 22.5801047910624930188688224465

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