Properties

Label 4-39e4-1.1-c3e2-0-5
Degree $4$
Conductor $2313441$
Sign $1$
Analytic cond. $8053.60$
Root an. cond. $9.47322$
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  − 16·4-s + 192·16-s − 250·25-s + 1.04e3·43-s + 286·49-s − 364·61-s − 2.04e3·64-s − 1.76e3·79-s + 4.00e3·100-s − 3.64e3·103-s − 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.66e4·172-s + 173-s + 179-s + 181-s + 191-s + 193-s − 4.57e3·196-s + ⋯
L(s)  = 1  − 2·4-s + 3·16-s − 2·25-s + 3.68·43-s + 0.833·49-s − 0.764·61-s − 4·64-s − 2.51·79-s + 4·100-s − 3.48·103-s − 2·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 7.37·172-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s − 1.66·196-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(8053.60\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1521} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2313441,\ (\ :3/2, 3/2),\ 1)\)

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 \( 1 \)
13 \( 1 \)
good2$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
5$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
7$C_2^2$ \( 1 - 286 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
17$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 10582 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 35282 T^{2} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 89206 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 520 T + p^{3} T^{2} )^{2} \)
47$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
53$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
59$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 182 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 172874 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 638066 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 + 884 T + p^{3} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
89$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 56446 T^{2} + p^{6} T^{4} \)
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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

−8.998214003713661364437147075987, −8.490328606881175717517447574420, −8.216853897304774644156514369066, −7.83327389115685696044828067506, −7.28937037436413480601284473953, −7.21079640178592173931393431180, −6.06808775661276535288782590172, −6.01113100590954679548542440306, −5.57895385905292561629522994632, −5.21533422105478407613540508452, −4.47156567398099798176236753724, −4.33717937869256377580484571026, −3.89040580809538059225491106695, −3.55505660731407155280105406811, −2.77187807113650451896034189970, −2.30600523739104581527236488882, −1.31732405888759754477880926371, −1.00655724279934310474327491070, 0, 0, 1.00655724279934310474327491070, 1.31732405888759754477880926371, 2.30600523739104581527236488882, 2.77187807113650451896034189970, 3.55505660731407155280105406811, 3.89040580809538059225491106695, 4.33717937869256377580484571026, 4.47156567398099798176236753724, 5.21533422105478407613540508452, 5.57895385905292561629522994632, 6.01113100590954679548542440306, 6.06808775661276535288782590172, 7.21079640178592173931393431180, 7.28937037436413480601284473953, 7.83327389115685696044828067506, 8.216853897304774644156514369066, 8.490328606881175717517447574420, 8.998214003713661364437147075987

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