Properties

Label 4-2736e2-1.1-c0e2-0-1
Degree $4$
Conductor $7485696$
Sign $1$
Analytic cond. $1.86443$
Root an. cond. $1.16852$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 13-s − 2·19-s + 25-s − 2·37-s − 3·43-s − 49-s + 61-s + 3·67-s − 73-s + 3·79-s + 2·97-s + 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 13-s − 2·19-s + 25-s − 2·37-s − 3·43-s − 49-s + 61-s + 3·67-s − 73-s + 3·79-s + 2·97-s + 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7485696\)    =    \(2^{8} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1.86443\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2736} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7485696,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9022563762\)
\(L(\frac12)\) \(\approx\) \(0.9022563762\)
\(L(1)\) 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 \( 1 \)
3 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 - T^{2} + T^{4} \)
7$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
17$C_2^2$ \( 1 - T^{2} + T^{4} \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_2^2$ \( 1 - T^{2} + T^{4} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 + T + T^{2} )^{2} \)
41$C_2^2$ \( 1 - T^{2} + T^{4} \)
43$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_2^2$ \( 1 - T^{2} + T^{4} \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2^2$ \( 1 - T^{2} + T^{4} \)
97$C_2$ \( ( 1 - T + T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.956347155768019861844335080888, −8.717835917660286676892196492538, −8.652040467403177403114760016448, −7.952547732500360601660244119708, −7.899988869357729301837164397719, −7.22247672072157391668508463573, −6.73317677564875107917777856273, −6.61629568662586173017187274181, −6.42056799225751556685081445375, −5.65695881297572421585454867291, −5.19079921750167853075494526374, −4.81377805780149879784234951135, −4.77565610126035143931323831375, −3.97943034627320081865768966328, −3.52801838285196907328727236870, −3.25102498689477295501650107012, −2.52484782754430229248165302089, −1.94496749112244977968865216055, −1.81980078647027655032016534291, −0.58543185996533308917789117660, 0.58543185996533308917789117660, 1.81980078647027655032016534291, 1.94496749112244977968865216055, 2.52484782754430229248165302089, 3.25102498689477295501650107012, 3.52801838285196907328727236870, 3.97943034627320081865768966328, 4.77565610126035143931323831375, 4.81377805780149879784234951135, 5.19079921750167853075494526374, 5.65695881297572421585454867291, 6.42056799225751556685081445375, 6.61629568662586173017187274181, 6.73317677564875107917777856273, 7.22247672072157391668508463573, 7.899988869357729301837164397719, 7.952547732500360601660244119708, 8.652040467403177403114760016448, 8.717835917660286676892196492538, 8.956347155768019861844335080888

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