Properties

Label 4-16640-1.1-c1e2-0-1
Degree $4$
Conductor $16640$
Sign $1$
Analytic cond. $1.06098$
Root an. cond. $1.01490$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 2·9-s + 3·13-s − 8·17-s + 2·25-s − 12·29-s − 12·37-s + 8·41-s + 6·45-s + 6·49-s + 20·53-s − 4·61-s + 9·65-s + 4·73-s − 5·81-s − 24·85-s − 4·89-s − 16·97-s + 12·101-s + 8·109-s + 6·117-s − 14·121-s − 10·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1.34·5-s + 2/3·9-s + 0.832·13-s − 1.94·17-s + 2/5·25-s − 2.22·29-s − 1.97·37-s + 1.24·41-s + 0.894·45-s + 6/7·49-s + 2.74·53-s − 0.512·61-s + 1.11·65-s + 0.468·73-s − 5/9·81-s − 2.60·85-s − 0.423·89-s − 1.62·97-s + 1.19·101-s + 0.766·109-s + 0.554·117-s − 1.27·121-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16640 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16640 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16640\)    =    \(2^{8} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(1.06098\)
Root analytic conductor: \(1.01490\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16640,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.373707389\)
\(L(\frac12)\) \(\approx\) \(1.373707389\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−10.99665603084136933695777268158, −10.43829028140359191111677736213, −10.03940461951279911716435138892, −9.253044758571529078077564771140, −9.025900933301165468584956072423, −8.525701822214155923531559284537, −7.52200466865104385187389079926, −7.00670577049561146871013475993, −6.46357626140388597733074399726, −5.72262240981459708464889979455, −5.37694307459617484012203042109, −4.29399920891880282775072036850, −3.76502980897935895002989580605, −2.38569504525986996554649613386, −1.73316942140065786408528891368, 1.73316942140065786408528891368, 2.38569504525986996554649613386, 3.76502980897935895002989580605, 4.29399920891880282775072036850, 5.37694307459617484012203042109, 5.72262240981459708464889979455, 6.46357626140388597733074399726, 7.00670577049561146871013475993, 7.52200466865104385187389079926, 8.525701822214155923531559284537, 9.025900933301165468584956072423, 9.253044758571529078077564771140, 10.03940461951279911716435138892, 10.43829028140359191111677736213, 10.99665603084136933695777268158

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