Properties

Label 4-33e4-1.1-c3e2-0-9
Degree $4$
Conductor $1185921$
Sign $1$
Analytic cond. $4128.45$
Root an. cond. $8.01580$
Motivic weight $3$
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  + 11·4-s − 18·5-s + 57·16-s − 198·20-s − 180·23-s − 7·25-s − 376·31-s + 266·37-s − 144·47-s − 98·49-s + 90·53-s − 756·59-s − 77·64-s − 772·67-s + 396·71-s − 1.02e3·80-s − 90·89-s − 1.98e3·92-s + 178·97-s − 77·100-s + 2.71e3·103-s − 1.42e3·113-s + 3.24e3·115-s − 4.13e3·124-s + 3.83e3·125-s + 127-s + 131-s + ⋯
L(s)  = 1  + 11/8·4-s − 1.60·5-s + 0.890·16-s − 2.21·20-s − 1.63·23-s − 0.0559·25-s − 2.17·31-s + 1.18·37-s − 0.446·47-s − 2/7·49-s + 0.233·53-s − 1.66·59-s − 0.150·64-s − 1.40·67-s + 0.661·71-s − 1.43·80-s − 0.107·89-s − 2.24·92-s + 0.186·97-s − 0.0769·100-s + 2.59·103-s − 1.18·113-s + 2.62·115-s − 2.99·124-s + 2.74·125-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1185921\)    =    \(3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(4128.45\)
Root analytic conductor: \(8.01580\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1185921,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
11 \( 1 \)
good2$C_2^2$ \( 1 - 11 T^{2} + p^{6} T^{4} \)
5$C_2$ \( ( 1 + 9 T + p^{3} T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 p^{2} T^{2} + p^{6} T^{4} \)
13$C_2^2$ \( 1 - 649 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 7 p^{2} T^{2} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 7450 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 + 90 T + p^{3} T^{2} )^{2} \)
29$C_2^2$ \( 1 + 40975 T^{2} + p^{6} T^{4} \)
31$C_2$ \( ( 1 + 188 T + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 133 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 136519 T^{2} + p^{6} T^{4} \)
43$C_2^2$ \( 1 + 153722 T^{2} + p^{6} T^{4} \)
47$C_2$ \( ( 1 + 72 T + p^{3} T^{2} )^{2} \)
53$C_2$ \( ( 1 - 45 T + p^{3} T^{2} )^{2} \)
59$C_2$ \( ( 1 + 378 T + p^{3} T^{2} )^{2} \)
61$C_2^2$ \( 1 + 65162 T^{2} + p^{6} T^{4} \)
67$C_2$ \( ( 1 + 386 T + p^{3} T^{2} )^{2} \)
71$C_2$ \( ( 1 - 198 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 772226 T^{2} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 962846 T^{2} + p^{6} T^{4} \)
83$C_2^2$ \( 1 - 411626 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 45 T + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 89 T + p^{3} T^{2} )^{2} \)
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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

−9.168242950203991871831383684096, −8.962392462629664165162762493397, −8.062765800811251700320273503980, −8.007563073803696298667191800203, −7.59884238186714628150922456234, −7.39492203841073370192048441493, −6.82621928452413478268042203128, −6.41044992771643547974800230149, −5.82647576282541233308808360260, −5.72676131280615909253419027207, −4.84449317060603320844272764206, −4.28523970700758217490435899618, −3.97500003521950425655114841536, −3.37910566208498455015516457589, −3.10573343945398107713669753124, −2.13316797235437596218135857267, −1.99494713538126959699582371801, −1.15747541587775228975876494663, 0, 0, 1.15747541587775228975876494663, 1.99494713538126959699582371801, 2.13316797235437596218135857267, 3.10573343945398107713669753124, 3.37910566208498455015516457589, 3.97500003521950425655114841536, 4.28523970700758217490435899618, 4.84449317060603320844272764206, 5.72676131280615909253419027207, 5.82647576282541233308808360260, 6.41044992771643547974800230149, 6.82621928452413478268042203128, 7.39492203841073370192048441493, 7.59884238186714628150922456234, 8.007563073803696298667191800203, 8.062765800811251700320273503980, 8.962392462629664165162762493397, 9.168242950203991871831383684096

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