Properties

Label 2-68544-1.1-c1-0-5
Degree $2$
Conductor $68544$
Sign $1$
Analytic cond. $547.326$
Root an. cond. $23.3950$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 7-s + 6·13-s − 17-s + 8·23-s − 25-s − 6·29-s − 8·31-s + 2·35-s − 10·37-s + 6·41-s − 12·43-s + 49-s − 10·53-s − 8·59-s − 6·61-s − 12·65-s − 12·67-s − 6·73-s − 8·79-s + 16·83-s + 2·85-s − 2·89-s − 6·91-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s + 1.66·13-s − 0.242·17-s + 1.66·23-s − 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.338·35-s − 1.64·37-s + 0.937·41-s − 1.82·43-s + 1/7·49-s − 1.37·53-s − 1.04·59-s − 0.768·61-s − 1.48·65-s − 1.46·67-s − 0.702·73-s − 0.900·79-s + 1.75·83-s + 0.216·85-s − 0.211·89-s − 0.628·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68544\)    =    \(2^{6} \cdot 3^{2} \cdot 7 \cdot 17\)
Sign: $1$
Analytic conductor: \(547.326\)
Root analytic conductor: \(23.3950\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{68544} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 68544,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9235043245\)
\(L(\frac12)\) \(\approx\) \(0.9235043245\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
17 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.09335388534929, −13.58347476143207, −13.11410578413344, −12.76102469550621, −12.17361351648651, −11.50216914962574, −11.16204317091138, −10.79816329605985, −10.26489529277113, −9.398050532906024, −8.981887672179233, −8.657411750583466, −7.995395982787787, −7.312637612065898, −7.126336712125865, −6.214694994207655, −5.961565238371375, −5.107753089486085, −4.622803611842555, −3.787976848584128, −3.439700058217476, −3.065364340000740, −1.851713470316598, −1.386097745818580, −0.3262037150475896, 0.3262037150475896, 1.386097745818580, 1.851713470316598, 3.065364340000740, 3.439700058217476, 3.787976848584128, 4.622803611842555, 5.107753089486085, 5.961565238371375, 6.214694994207655, 7.126336712125865, 7.312637612065898, 7.995395982787787, 8.657411750583466, 8.981887672179233, 9.398050532906024, 10.26489529277113, 10.79816329605985, 11.16204317091138, 11.50216914962574, 12.17361351648651, 12.76102469550621, 13.11410578413344, 13.58347476143207, 14.09335388534929

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