Properties

Label 4-1216e2-1.1-c2e2-0-6
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $1097.83$
Root an. cond. $5.75617$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9·5-s + 5·7-s + 18·9-s + 3·11-s − 15·17-s + 38·19-s − 60·23-s + 25·25-s − 45·35-s − 85·43-s − 162·45-s − 75·47-s + 49·49-s − 27·55-s + 103·61-s + 90·63-s + 25·73-s + 15·77-s + 243·81-s − 180·83-s + 135·85-s − 342·95-s + 54·99-s + 204·101-s + 540·115-s − 75·119-s + 121·121-s + ⋯
L(s)  = 1  − 9/5·5-s + 5/7·7-s + 2·9-s + 3/11·11-s − 0.882·17-s + 2·19-s − 2.60·23-s + 25-s − 9/7·35-s − 1.97·43-s − 3.59·45-s − 1.59·47-s + 49-s − 0.490·55-s + 1.68·61-s + 10/7·63-s + 0.342·73-s + 0.194·77-s + 3·81-s − 2.16·83-s + 1.58·85-s − 3.59·95-s + 6/11·99-s + 2.01·101-s + 4.69·115-s − 0.630·119-s + 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1097.83\)
Root analytic conductor: \(5.75617\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.421802517\)
\(L(\frac12)\) \(\approx\) \(1.421802517\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
19$C_1$ \( ( 1 - p T )^{2} \)
good3$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
5$C_2^2$ \( 1 + 9 T + 56 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2^2$ \( 1 - 5 T - 24 T^{2} - 5 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 112 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
17$C_2^2$ \( 1 + 15 T - 64 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} \)
23$C_2$ \( ( 1 + 30 T + p^{2} T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
43$C_2^2$ \( 1 + 85 T + 5376 T^{2} + 85 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2^2$ \( 1 + 75 T + 3416 T^{2} + 75 p^{2} T^{3} + p^{4} T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
61$C_2^2$ \( 1 - 103 T + 6888 T^{2} - 103 p^{2} T^{3} + p^{4} T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2^2$ \( 1 - 25 T - 4704 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
83$C_2$ \( ( 1 + 90 T + p^{2} T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.656161300041797426356212897973, −9.655673269666963791873207697637, −8.602405603414539888927232964383, −8.579596543178915337481251867210, −7.85320533508161270552148029886, −7.82037675583892196228682400844, −7.29001599711649008202969349806, −7.15938970065680280415891923807, −6.46260897176344204746417173242, −6.19187828276540075727027518694, −5.16723219614876756657440988069, −5.10892463903404308547963880694, −4.32363852995342092160394397940, −4.20389069497851144119118518502, −3.62831969914893079497934577170, −3.49134520444998288406440769099, −2.40414402109962889480599714349, −1.71954298559270567831099226396, −1.30036670126043883232048030226, −0.37696500066613395303721267576, 0.37696500066613395303721267576, 1.30036670126043883232048030226, 1.71954298559270567831099226396, 2.40414402109962889480599714349, 3.49134520444998288406440769099, 3.62831969914893079497934577170, 4.20389069497851144119118518502, 4.32363852995342092160394397940, 5.10892463903404308547963880694, 5.16723219614876756657440988069, 6.19187828276540075727027518694, 6.46260897176344204746417173242, 7.15938970065680280415891923807, 7.29001599711649008202969349806, 7.82037675583892196228682400844, 7.85320533508161270552148029886, 8.579596543178915337481251867210, 8.602405603414539888927232964383, 9.655673269666963791873207697637, 9.656161300041797426356212897973

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