Properties

Label 4-2736e2-1.1-c0e2-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(1.296867386\)
\(L(\frac12)\) \(\approx\) \(1.296867386\)
\(L(1)\) 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

−8.952910772874176106280888074193, −8.943994934304339261617236592804, −8.655110398597480460679181888037, −7.87273262613030846417016401448, −7.63823020565407308784191310789, −7.34073364062009789488295651085, −6.96516226400799493069668125762, −6.77888406630811769064961417941, −5.83912396319886297117849941603, −5.79609977516594648501204724341, −5.46945916640562172902027589448, −4.70815618028868880075134397474, −4.66579629539841551547516813598, −4.17810278864933120319254758079, −3.31808992649296495843495233733, −3.18985694617474133857687924156, −2.76213216514709892984274457454, −2.05880563793082294983593070623, −1.50606362395437365325097187522, −0.76959534355878156740201573646, 0.76959534355878156740201573646, 1.50606362395437365325097187522, 2.05880563793082294983593070623, 2.76213216514709892984274457454, 3.18985694617474133857687924156, 3.31808992649296495843495233733, 4.17810278864933120319254758079, 4.66579629539841551547516813598, 4.70815618028868880075134397474, 5.46945916640562172902027589448, 5.79609977516594648501204724341, 5.83912396319886297117849941603, 6.77888406630811769064961417941, 6.96516226400799493069668125762, 7.34073364062009789488295651085, 7.63823020565407308784191310789, 7.87273262613030846417016401448, 8.655110398597480460679181888037, 8.943994934304339261617236592804, 8.952910772874176106280888074193

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