Properties

Label 4-1008e2-1.1-c1e2-0-7
Degree $4$
Conductor $1016064$
Sign $1$
Analytic cond. $64.7851$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 7-s + 3·11-s − 6·17-s − 2·19-s + 25-s − 18·29-s + 5·31-s + 3·35-s − 10·37-s − 12·47-s − 6·49-s − 9·53-s − 9·55-s + 9·59-s + 24·67-s − 12·73-s − 3·77-s − 9·79-s + 6·83-s + 18·85-s + 6·89-s + 6·95-s + 24·101-s − 4·103-s − 21·107-s + 4·109-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.377·7-s + 0.904·11-s − 1.45·17-s − 0.458·19-s + 1/5·25-s − 3.34·29-s + 0.898·31-s + 0.507·35-s − 1.64·37-s − 1.75·47-s − 6/7·49-s − 1.23·53-s − 1.21·55-s + 1.17·59-s + 2.93·67-s − 1.40·73-s − 0.341·77-s − 1.01·79-s + 0.658·83-s + 1.95·85-s + 0.635·89-s + 0.615·95-s + 2.38·101-s − 0.394·103-s − 2.03·107-s + 0.383·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(64.7851\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1008} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1016064,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3788361084\)
\(L(\frac12)\) \(\approx\) \(0.3788361084\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 24 T + 259 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T + 101 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−10.17209322392083371939006883846, −9.621331715903732146657447270145, −9.334260454576718825189320708617, −8.938066155308715120634578520773, −8.510660708874517431189346555145, −8.060516017578463709233151869503, −7.74652909162801665273434232794, −7.20535915585831348465217385630, −6.78578109995896147618374108247, −6.50156276903900781543605011615, −6.05483228911911422583028551978, −5.23477944837146269916491119455, −5.02891482082621981463035529352, −4.08381851655514079750408090954, −4.07088848979079801029221540426, −3.56551130394355674748256170465, −3.04134883912938463632150404907, −2.05808836085383305465113331015, −1.67417364427564755393770546381, −0.27374820706509433817429088343, 0.27374820706509433817429088343, 1.67417364427564755393770546381, 2.05808836085383305465113331015, 3.04134883912938463632150404907, 3.56551130394355674748256170465, 4.07088848979079801029221540426, 4.08381851655514079750408090954, 5.02891482082621981463035529352, 5.23477944837146269916491119455, 6.05483228911911422583028551978, 6.50156276903900781543605011615, 6.78578109995896147618374108247, 7.20535915585831348465217385630, 7.74652909162801665273434232794, 8.060516017578463709233151869503, 8.510660708874517431189346555145, 8.938066155308715120634578520773, 9.334260454576718825189320708617, 9.621331715903732146657447270145, 10.17209322392083371939006883846

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