Properties

Label 4-80e4-1.1-c1e2-0-30
Degree $4$
Conductor $40960000$
Sign $1$
Analytic cond. $2611.64$
Root an. cond. $7.14872$
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  − 8·17-s − 16·29-s + 8·37-s − 16·41-s − 8·49-s + 16·53-s − 12·61-s − 8·73-s − 9·81-s + 4·89-s + 8·97-s − 16·101-s + 20·109-s + 32·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1.94·17-s − 2.97·29-s + 1.31·37-s − 2.49·41-s − 8/7·49-s + 2.19·53-s − 1.53·61-s − 0.936·73-s − 81-s + 0.423·89-s + 0.812·97-s − 1.59·101-s + 1.91·109-s + 3.01·113-s + 2/11·121-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 − 2·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40960000\)    =    \(2^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2611.64\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 40960000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 160 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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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

−7.71832627834747909288774991947, −7.56022923673304581525121920605, −6.98804009558510783854578858897, −6.97929518577251570045348602820, −6.45966312445147218822801253650, −6.07195532910014085295109807669, −5.70104471382508399234376644509, −5.47323264744782359453053984863, −4.91447937823470135637597502559, −4.45454894691241859383511689586, −4.40437736230118030041057856727, −3.79016187240491880953291569848, −3.35717223246557514948188935694, −3.15089600182070909283723960231, −2.26040715267753578253977073249, −2.18628486065288014965636131715, −1.71145543516659841251904292902, −1.07845093473388407883968326075, 0, 0, 1.07845093473388407883968326075, 1.71145543516659841251904292902, 2.18628486065288014965636131715, 2.26040715267753578253977073249, 3.15089600182070909283723960231, 3.35717223246557514948188935694, 3.79016187240491880953291569848, 4.40437736230118030041057856727, 4.45454894691241859383511689586, 4.91447937823470135637597502559, 5.47323264744782359453053984863, 5.70104471382508399234376644509, 6.07195532910014085295109807669, 6.45966312445147218822801253650, 6.97929518577251570045348602820, 6.98804009558510783854578858897, 7.56022923673304581525121920605, 7.71832627834747909288774991947

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