Properties

Label 4-1008e2-1.1-c5e2-0-14
Degree $4$
Conductor $1016064$
Sign $1$
Analytic cond. $26136.1$
Root an. cond. $12.7148$
Motivic weight $5$
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  − 80·5-s − 98·7-s + 432·11-s − 876·13-s + 848·17-s + 3.35e3·19-s − 5.29e3·23-s + 266·25-s + 256·29-s + 856·31-s + 7.84e3·35-s + 3.63e3·37-s + 3.05e3·41-s + 2.62e4·43-s + 2.91e3·47-s + 7.20e3·49-s + 2.72e4·53-s − 3.45e4·55-s + 5.90e4·59-s − 4.04e4·61-s + 7.00e4·65-s − 5.29e4·67-s − 5.27e4·71-s + 1.88e4·73-s − 4.23e4·77-s − 5.02e4·79-s − 6.60e4·83-s + ⋯
L(s)  = 1  − 1.43·5-s − 0.755·7-s + 1.07·11-s − 1.43·13-s + 0.711·17-s + 2.13·19-s − 2.08·23-s + 0.0851·25-s + 0.0565·29-s + 0.159·31-s + 1.08·35-s + 0.436·37-s + 0.283·41-s + 2.16·43-s + 0.192·47-s + 3/7·49-s + 1.33·53-s − 1.54·55-s + 2.20·59-s − 1.39·61-s + 2.05·65-s − 1.44·67-s − 1.24·71-s + 0.413·73-s − 0.813·77-s − 0.906·79-s − 1.05·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(26136.1\)
Root analytic conductor: \(12.7148\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1016064,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 + p^{2} T )^{2} \)
good5$D_{4}$ \( 1 + 16 p T + 6134 T^{2} + 16 p^{6} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 432 T + 353314 T^{2} - 432 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 876 T + 495134 T^{2} + 876 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 848 T + 2633390 T^{2} - 848 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 3352 T + 385362 p T^{2} - 3352 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 5296 T + 19882874 T^{2} + 5296 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 256 T - 19611622 T^{2} - 256 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 856 T - 16799538 T^{2} - 856 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 3636 T + 88838222 T^{2} - 3636 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 3056 T - 47419714 T^{2} - 3056 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 26216 T + 379734534 T^{2} - 26216 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 2912 T + 449271566 T^{2} - 2912 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 27232 T + 861107066 T^{2} - 27232 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 59040 T + 2034894022 T^{2} - 59040 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 40420 T + 1559499102 T^{2} + 40420 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 52920 T + 3287922038 T^{2} + 52920 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 52752 T + 1754243002 T^{2} + 52752 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 18812 T + 676318422 T^{2} - 18812 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 50288 T + 5505346398 T^{2} + 50288 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 66048 T + 8599217926 T^{2} + 66048 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 131504 T + 14283515966 T^{2} - 131504 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 164348 T + 23630282694 T^{2} - 164348 p^{5} T^{3} + p^{10} T^{4} \)
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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

−9.065177669357696759746049466297, −8.691081174672342689387434617666, −7.922709120305317669816824655712, −7.68866146925080647888815721634, −7.44207592359686628674241754045, −7.18795758970290832053440849532, −6.52401463942860405473637564265, −5.87829770829742679621547931948, −5.77607484198374259979852477270, −5.12616711312231625208631785987, −4.37740730790668862888850354649, −4.20204117472260270830576028129, −3.55136128775830687635339455942, −3.47627810818129741369723508827, −2.56138344703063950124839255276, −2.33923020940238225144379530886, −1.21363350448213548046294765689, −0.958003425478879581946950340638, 0, 0, 0.958003425478879581946950340638, 1.21363350448213548046294765689, 2.33923020940238225144379530886, 2.56138344703063950124839255276, 3.47627810818129741369723508827, 3.55136128775830687635339455942, 4.20204117472260270830576028129, 4.37740730790668862888850354649, 5.12616711312231625208631785987, 5.77607484198374259979852477270, 5.87829770829742679621547931948, 6.52401463942860405473637564265, 7.18795758970290832053440849532, 7.44207592359686628674241754045, 7.68866146925080647888815721634, 7.922709120305317669816824655712, 8.691081174672342689387434617666, 9.065177669357696759746049466297

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