Properties

Label 4-790272-1.1-c1e2-0-25
Degree $4$
Conductor $790272$
Sign $1$
Analytic cond. $50.3884$
Root an. cond. $2.66429$
Motivic weight $1$
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  + 2·3-s − 7-s + 9-s + 12·17-s − 2·21-s − 10·25-s − 4·27-s + 4·37-s + 12·41-s − 16·43-s + 24·47-s + 49-s + 24·51-s + 12·59-s − 63-s + 8·67-s − 20·75-s − 16·79-s − 11·81-s + 12·83-s − 12·89-s + 4·109-s + 8·111-s − 12·119-s − 22·121-s + 24·123-s + 127-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.377·7-s + 1/3·9-s + 2.91·17-s − 0.436·21-s − 2·25-s − 0.769·27-s + 0.657·37-s + 1.87·41-s − 2.43·43-s + 3.50·47-s + 1/7·49-s + 3.36·51-s + 1.56·59-s − 0.125·63-s + 0.977·67-s − 2.30·75-s − 1.80·79-s − 1.22·81-s + 1.31·83-s − 1.27·89-s + 0.383·109-s + 0.759·111-s − 1.10·119-s − 2·121-s + 2.16·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(790272\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(50.3884\)
Root analytic conductor: \(2.66429\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 790272,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.067143272\)
\(L(\frac12)\) \(\approx\) \(3.067143272\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.182643116839158569055135937831, −7.88115201382147028507134927026, −7.39293674981761532961441986375, −7.29668061464756137953129595214, −6.43902889849944115155691598253, −5.83259812621372959122100845675, −5.62168100149868099368912779374, −5.25590580602395553760699468065, −4.24465289176609138955904372614, −3.82418745638366191622617161116, −3.56550924639142318669266713404, −2.84501304641458537647904209821, −2.48514181668809031419660323366, −1.67921161671410140538744564928, −0.818365395076437722978448101852, 0.818365395076437722978448101852, 1.67921161671410140538744564928, 2.48514181668809031419660323366, 2.84501304641458537647904209821, 3.56550924639142318669266713404, 3.82418745638366191622617161116, 4.24465289176609138955904372614, 5.25590580602395553760699468065, 5.62168100149868099368912779374, 5.83259812621372959122100845675, 6.43902889849944115155691598253, 7.29668061464756137953129595214, 7.39293674981761532961441986375, 7.88115201382147028507134927026, 8.182643116839158569055135937831

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