Properties

Label 4-336e2-1.1-c5e2-0-1
Degree $4$
Conductor $112896$
Sign $1$
Analytic cond. $2904.02$
Root an. cond. $7.34091$
Motivic weight $5$
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  − 9·3-s − 86·5-s − 49·7-s + 34·11-s − 6·13-s + 774·15-s + 1.90e3·17-s − 1.48e3·19-s + 441·21-s − 224·23-s + 3.12e3·25-s + 729·27-s − 1.30e4·29-s + 1.73e3·31-s − 306·33-s + 4.21e3·35-s + 7.63e3·37-s + 54·39-s + 3.08e4·41-s − 3.69e4·43-s + 1.84e4·47-s − 1.44e4·49-s − 1.71e4·51-s + 1.99e4·53-s − 2.92e3·55-s + 1.34e4·57-s − 3.18e4·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.53·5-s − 0.377·7-s + 0.0847·11-s − 0.00984·13-s + 0.888·15-s + 1.59·17-s − 0.946·19-s + 0.218·21-s − 0.0882·23-s + 25-s + 0.192·27-s − 2.87·29-s + 0.323·31-s − 0.0489·33-s + 0.581·35-s + 0.916·37-s + 0.00568·39-s + 2.86·41-s − 3.05·43-s + 1.21·47-s − 6/7·49-s − 0.922·51-s + 0.975·53-s − 0.130·55-s + 0.546·57-s − 1.19·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(112896\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2904.02\)
Root analytic conductor: \(7.34091\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 112896,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.3489279847\)
\(L(\frac12)\) \(\approx\) \(0.3489279847\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 + p^{2} T + p^{4} T^{2} \)
7$C_2$ \( 1 + p^{2} T + p^{5} T^{2} \)
good5$C_2^2$ \( 1 + 86 T + 4271 T^{2} + 86 p^{5} T^{3} + p^{10} T^{4} \)
11$C_2^2$ \( 1 - 34 T - 159895 T^{2} - 34 p^{5} T^{3} + p^{10} T^{4} \)
13$C_2$ \( ( 1 + 3 T + p^{5} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 112 p T + 7631 p^{2} T^{2} - 112 p^{6} T^{3} + p^{10} T^{4} \)
19$C_2^2$ \( 1 + 1489 T - 258978 T^{2} + 1489 p^{5} T^{3} + p^{10} T^{4} \)
23$C_2^2$ \( 1 + 224 T - 6386167 T^{2} + 224 p^{5} T^{3} + p^{10} T^{4} \)
29$C_2$ \( ( 1 + 6508 T + p^{5} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 1731 T - 25632790 T^{2} - 1731 p^{5} T^{3} + p^{10} T^{4} \)
37$C_2^2$ \( 1 - 7633 T - 11081268 T^{2} - 7633 p^{5} T^{3} + p^{10} T^{4} \)
41$C_2$ \( ( 1 - 15414 T + p^{5} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 18491 T + p^{5} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 18462 T + 111500437 T^{2} - 18462 p^{5} T^{3} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 19956 T - 19953557 T^{2} - 19956 p^{5} T^{3} + p^{10} T^{4} \)
59$C_2^2$ \( 1 + 31828 T + 298097285 T^{2} + 31828 p^{5} T^{3} + p^{10} T^{4} \)
61$C_2^2$ \( 1 - 57654 T + 2479387415 T^{2} - 57654 p^{5} T^{3} + p^{10} T^{4} \)
67$C_2^2$ \( 1 + 60563 T + 2317751862 T^{2} + 60563 p^{5} T^{3} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 44834 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 20821 T - 1639557552 T^{2} + 20821 p^{5} T^{3} + p^{10} T^{4} \)
79$C_2^2$ \( 1 + 30531 T - 2144914438 T^{2} + 30531 p^{5} T^{3} + p^{10} T^{4} \)
83$C_2$ \( ( 1 + 110602 T + p^{5} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 58992 T - 2104003385 T^{2} - 58992 p^{5} T^{3} + p^{10} T^{4} \)
97$C_2$ \( ( 1 + 119846 T + p^{5} T^{2} )^{2} \)
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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

−11.34259995043504293287818801857, −10.70178923188129224659544950474, −9.830461930332676533824696367543, −9.817473108018282603088993141827, −9.112256544030239112704346259981, −8.417022581578570122374377556012, −8.145359758083870007747332474261, −7.52079214703007463964408120940, −7.29918099675725587713616994364, −6.72080118555812828400635126480, −5.82604611142126503147585123740, −5.79583603946354349110715348309, −5.03876576491856018071939204744, −4.13835196800899946721462848312, −4.05126388005971583712498279849, −3.36347477188803047367950057344, −2.74894879406046633554094174919, −1.77427920326629623305857874680, −0.934503268767898666191670251091, −0.19961444482186620196488331932, 0.19961444482186620196488331932, 0.934503268767898666191670251091, 1.77427920326629623305857874680, 2.74894879406046633554094174919, 3.36347477188803047367950057344, 4.05126388005971583712498279849, 4.13835196800899946721462848312, 5.03876576491856018071939204744, 5.79583603946354349110715348309, 5.82604611142126503147585123740, 6.72080118555812828400635126480, 7.29918099675725587713616994364, 7.52079214703007463964408120940, 8.145359758083870007747332474261, 8.417022581578570122374377556012, 9.112256544030239112704346259981, 9.817473108018282603088993141827, 9.830461930332676533824696367543, 10.70178923188129224659544950474, 11.34259995043504293287818801857

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