Properties

Label 4-18e4-1.1-c1e2-0-5
Degree $4$
Conductor $104976$
Sign $1$
Analytic cond. $6.69336$
Root an. cond. $1.60846$
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  + 4·7-s + 10·13-s + 4·19-s − 25-s − 8·31-s + 10·37-s − 20·43-s − 2·49-s + 10·61-s + 4·67-s − 2·73-s − 20·79-s + 40·91-s − 20·97-s − 32·103-s − 14·109-s + 14·121-s + 127-s + 131-s + 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 1.51·7-s + 2.77·13-s + 0.917·19-s − 1/5·25-s − 1.43·31-s + 1.64·37-s − 3.04·43-s − 2/7·49-s + 1.28·61-s + 0.488·67-s − 0.234·73-s − 2.25·79-s + 4.19·91-s − 2.03·97-s − 3.15·103-s − 1.34·109-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 1.38·133-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(104976\)    =    \(2^{4} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(6.69336\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 104976,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.123273647\)
\(L(\frac12)\) \(\approx\) \(2.123273647\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−9.677528525390612960317867289160, −8.776320410384522753279292613425, −8.579573961859898822258311157623, −7.957203241521051791394470373598, −7.949251102855060400602455037247, −6.89563277027074651963402661567, −6.64268335409264911701355422086, −5.72299699566267407044406059705, −5.57441037932249871861950405953, −4.85946862051001909635264517777, −4.13912411526585097190750656835, −3.66391361115582894073101159026, −2.97036565258433729627708570907, −1.66955489747042459625131055657, −1.33745094888120597007437567736, 1.33745094888120597007437567736, 1.66955489747042459625131055657, 2.97036565258433729627708570907, 3.66391361115582894073101159026, 4.13912411526585097190750656835, 4.85946862051001909635264517777, 5.57441037932249871861950405953, 5.72299699566267407044406059705, 6.64268335409264911701355422086, 6.89563277027074651963402661567, 7.949251102855060400602455037247, 7.957203241521051791394470373598, 8.579573961859898822258311157623, 8.776320410384522753279292613425, 9.677528525390612960317867289160

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