↑

Properties

Label 2-168e2-1.1-c1-0-102
Degree $2$
Conductor $28224$
Sign $-1$
Analytic cond. $225.369$
Root an. cond. $15.0123$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·11-s + 2·17-s + 2·23-s − 25-s + 6·29-s − 4·31-s − 6·37-s − 2·41-s − 6·53-s − 4·55-s + 12·59-s + 12·61-s − 12·67-s − 10·71-s − 12·73-s − 12·79-s + 12·83-s − 4·85-s + 14·89-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 4·115-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.603·11-s + 0.485·17-s + 0.417·23-s − 1/5·25-s + 1.11·29-s − 0.718·31-s − 0.986·37-s − 0.312·41-s − 0.824·53-s − 0.539·55-s + 1.56·59-s + 1.53·61-s − 1.46·67-s − 1.18·71-s − 1.40·73-s − 1.35·79-s + 1.31·83-s − 0.433·85-s + 1.48·89-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.373·115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28224\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(225.369\)
Root analytic conductor: \(15.0123\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 28224,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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

−15.58869695726202, −14.82192315568652, −14.55427450554206, −14.00124263970016, −13.25300389153837, −12.85468032802147, −12.06283692990750, −11.78981144325315, −11.39437626826948, −10.54164811506323, −10.21248716195394, −9.484555319236007, −8.759868193940832, −8.501578911313144, −7.663980772476476, −7.299724434299986, −6.650153694380262, −6.024720427939474, −5.272381892889857, −4.669904900392032, −3.955550644697152, −3.483268903477003, −2.786614598218516, −1.799815452828849, −0.9907036364383752, 0, 0.9907036364383752, 1.799815452828849, 2.786614598218516, 3.483268903477003, 3.955550644697152, 4.669904900392032, 5.272381892889857, 6.024720427939474, 6.650153694380262, 7.299724434299986, 7.663980772476476, 8.501578911313144, 8.759868193940832, 9.484555319236007, 10.21248716195394, 10.54164811506323, 11.39437626826948, 11.78981144325315, 12.06283692990750, 12.85468032802147, 13.25300389153837, 14.00124263970016, 14.55427450554206, 14.82192315568652, 15.58869695726202

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