Properties

Label 4-6e8-1.1-c1e2-0-41
Degree $4$
Conductor $1679616$
Sign $1$
Analytic cond. $107.093$
Root an. cond. $3.21692$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·7-s − 7·13-s − 5·25-s − 18·31-s − 2·37-s − 18·43-s − 49-s + 13·61-s − 21·67-s − 34·73-s + 21·79-s + 21·91-s + 5·97-s − 33·103-s − 4·109-s + 11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 13·169-s + 173-s + ⋯
L(s)  = 1  − 1.13·7-s − 1.94·13-s − 25-s − 3.23·31-s − 0.328·37-s − 2.74·43-s − 1/7·49-s + 1.66·61-s − 2.56·67-s − 3.97·73-s + 2.36·79-s + 2.20·91-s + 0.507·97-s − 3.25·103-s − 0.383·109-s + 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 + 169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1679616\)    =    \(2^{8} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(107.093\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1679616,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
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$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−9.367442730773255774982940818209, −9.318200144289380673419647218715, −8.697123802597151813263439394246, −8.276757127874665655284573368061, −7.63190557378279379830316841925, −7.44002287338258914107594420830, −6.86412660394563762024659658038, −6.85897222873525122875098841619, −6.05182093103421918018518510515, −5.70186709727430272841877203075, −5.21800233738994317070411948060, −4.89764861175858715021026909054, −4.19968525243674936408754935089, −3.76384018313927621056478477213, −3.16206579279057675422208939274, −2.88555062163160274369068024720, −1.97807547862772260725466015185, −1.70950401533280722758584861901, 0, 0, 1.70950401533280722758584861901, 1.97807547862772260725466015185, 2.88555062163160274369068024720, 3.16206579279057675422208939274, 3.76384018313927621056478477213, 4.19968525243674936408754935089, 4.89764861175858715021026909054, 5.21800233738994317070411948060, 5.70186709727430272841877203075, 6.05182093103421918018518510515, 6.85897222873525122875098841619, 6.86412660394563762024659658038, 7.44002287338258914107594420830, 7.63190557378279379830316841925, 8.276757127874665655284573368061, 8.697123802597151813263439394246, 9.318200144289380673419647218715, 9.367442730773255774982940818209

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