Properties

Label 4-4050e2-1.1-c1e2-0-10
Degree $4$
Conductor $16402500$
Sign $1$
Analytic cond. $1045.83$
Root an. cond. $5.68677$
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-s − 6·11-s + 16-s + 2·19-s − 12·29-s − 8·31-s + 18·41-s + 6·44-s + 10·49-s − 6·59-s + 16·61-s − 64-s − 24·71-s − 2·76-s + 8·79-s − 12·89-s + 32·109-s + 12·116-s + 5·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.80·11-s + 1/4·16-s + 0.458·19-s − 2.22·29-s − 1.43·31-s + 2.81·41-s + 0.904·44-s + 10/7·49-s − 0.781·59-s + 2.04·61-s − 1/8·64-s − 2.84·71-s − 0.229·76-s + 0.900·79-s − 1.27·89-s + 3.06·109-s + 1.11·116-s + 5/11·121-s + 0.718·124-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16402500\)    =    \(2^{2} \cdot 3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1045.83\)
Root analytic conductor: \(5.68677\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16402500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.045839442\)
\(L(\frac12)\) \(\approx\) \(1.045839442\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 169 T^{2} + p^{2} T^{4} \)
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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

−8.573655929525949194277979291149, −8.288613093056525379853628318085, −7.77495190820512617759022568745, −7.50234349054099782658974287395, −7.20324062130633471403189291504, −7.19144141109860481872835485648, −6.20413514131693861982041490558, −5.91616301540213038691580802429, −5.54535007371167780100095785487, −5.43518932706930470617349690036, −4.94574398990925843426805092346, −4.45017539345972887128022795222, −3.97550133974846751462914862583, −3.80151170793588899759152225065, −2.95084280559961219696300970819, −2.93668651995665835282059551390, −2.09006320725051086696292629093, −1.94220815929364722072623333440, −0.972686667957262494455480185444, −0.34128261144478683919670392729, 0.34128261144478683919670392729, 0.972686667957262494455480185444, 1.94220815929364722072623333440, 2.09006320725051086696292629093, 2.93668651995665835282059551390, 2.95084280559961219696300970819, 3.80151170793588899759152225065, 3.97550133974846751462914862583, 4.45017539345972887128022795222, 4.94574398990925843426805092346, 5.43518932706930470617349690036, 5.54535007371167780100095785487, 5.91616301540213038691580802429, 6.20413514131693861982041490558, 7.19144141109860481872835485648, 7.20324062130633471403189291504, 7.50234349054099782658974287395, 7.77495190820512617759022568745, 8.288613093056525379853628318085, 8.573655929525949194277979291149

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