Properties

Label 4-2736e2-1.1-c1e2-0-17
Degree $4$
Conductor $7485696$
Sign $1$
Analytic cond. $477.294$
Root an. cond. $4.67408$
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  − 3·13-s + 8·19-s + 5·25-s + 22·31-s − 21·43-s − 13·49-s − 13·61-s − 5·67-s − 7·73-s − 13·79-s − 24·97-s + 26·103-s − 36·109-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 7·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 0.832·13-s + 1.83·19-s + 25-s + 3.95·31-s − 3.20·43-s − 1.85·49-s − 1.66·61-s − 0.610·67-s − 0.819·73-s − 1.46·79-s − 2.43·97-s + 2.56·103-s − 3.44·109-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.538·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(7485696\)    =    \(2^{8} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(477.294\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2736} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7485696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.028439625\)
\(L(\frac12)\) \(\approx\) \(2.028439625\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−8.953445377401040021498522917011, −8.477167174427032942090309621137, −8.387449148627503355571768811530, −7.79333817988902189197563936966, −7.63045220819182003893668740789, −7.11757346460937024290737439873, −6.54803589164231758125137293313, −6.54465883357196843156794096547, −6.07898805639116292969712810577, −5.31379961824628724112932444678, −5.15882006201026845946440081859, −4.62908986657870710216832762363, −4.56410261551831355912584166795, −3.80898054312251139821562709701, −3.05282148851518485949227202947, −2.96249654476856726277418282109, −2.68949603577665229247403850346, −1.51493799715985035934529109271, −1.42848672925635331159848120899, −0.47951646416331324907246576272, 0.47951646416331324907246576272, 1.42848672925635331159848120899, 1.51493799715985035934529109271, 2.68949603577665229247403850346, 2.96249654476856726277418282109, 3.05282148851518485949227202947, 3.80898054312251139821562709701, 4.56410261551831355912584166795, 4.62908986657870710216832762363, 5.15882006201026845946440081859, 5.31379961824628724112932444678, 6.07898805639116292969712810577, 6.54465883357196843156794096547, 6.54803589164231758125137293313, 7.11757346460937024290737439873, 7.63045220819182003893668740789, 7.79333817988902189197563936966, 8.387449148627503355571768811530, 8.477167174427032942090309621137, 8.953445377401040021498522917011

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