↑

Properties

Label 1-684-684.7-r1-0-0
Degree $1$
Conductor $684$
Sign $-0.612 + 0.790i$
Analytic cond. $73.5060$
Root an. cond. $73.5060$
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  + (−0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)11-s + 13-s + (−0.5 + 0.866i)17-s − 23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + (0.5 − 0.866i)35-s + 37-s + (−0.5 − 0.866i)41-s − 43-s + (0.5 − 0.866i)47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)11-s + 13-s + (−0.5 + 0.866i)17-s − 23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + (0.5 − 0.866i)35-s + 37-s + (−0.5 − 0.866i)41-s − 43-s + (0.5 − 0.866i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.612 + 0.790i$
Analytic conductor: \(73.5060\)
Root analytic conductor: \(73.5060\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 684,\ (1:\ ),\ -0.612 + 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4652179069 + 0.9497203697i\)
\(L(\frac12)\) \(\approx\) \(0.4652179069 + 0.9497203697i\)
\(L(1)\) \(\approx\) \(0.9520732537 + 0.1304924376i\)
\(L(1)\) \(\approx\) \(0.9520732537 + 0.1304924376i\)

Euler product

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

−22.24872409702095712945608019879, −21.50912421550900512847898671249, −20.442922407211345381536182190152, −19.86326100007290333429162662215, −18.86499407089081207058646680581, −18.22862887415498938342275531478, −17.387341634663840602898214664881, −16.338235588602633006279025408287, −15.71884497857749009828992885615, −14.65070722455885946096363785154, −13.91018282852954500426786885235, −13.3936133646610207136212068316, −11.85283468187538920820019899115, −11.27125013343064806735345233687, −10.65864819224832267288927235441, −9.62165545820781334748229219780, −8.39522642387881069681011696888, −7.742690138568204559998140531445, −6.71179954769402979704499075160, −6.03411793982560573962229054876, −4.55198281785951868216343952044, −3.77279101450862072624977353153, −2.88756619423169181107600704892, −1.43705666755293072243045534712, −0.25316403129438905352807719533, 1.320706657486610530229449281906, 2.12007362011317697400131512822, 3.729495083221605419869464402064, 4.43938125326048091966227610410, 5.46648756089264640699843158031, 6.34335619343912600219603241438, 7.61030724987534090295062631512, 8.5048057962441430100578679887, 8.99038167276209530535282953567, 10.1058898073545391183828173358, 11.31700260966248476355496123559, 11.93267430450613958119182521612, 12.712393228714975285117715207368, 13.52685815299721765520446707873, 14.80405353626544701641995369714, 15.31832517628245612347489238908, 16.17565383386655347303971811250, 17.05242537200231466795904411337, 17.92031853270486501742041020694, 18.649120806145621226493496171726, 19.72123300078903176696744518870, 20.32198582324872096113659866175, 21.093575442741429277810322052570, 21.96274838637191337706295906738, 22.78993415887085817648966085267

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