Properties

Label 4-39e4-1.1-c1e2-0-9
Degree $4$
Conductor $2313441$
Sign $1$
Analytic cond. $147.507$
Root an. cond. $3.48500$
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  + 3·4-s + 5·16-s + 4·17-s + 6·25-s + 20·29-s + 24·43-s − 2·49-s − 12·53-s − 4·61-s + 3·64-s + 12·68-s + 16·79-s + 18·100-s − 36·101-s − 24·107-s + 12·113-s + 60·116-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 3/2·4-s + 5/4·16-s + 0.970·17-s + 6/5·25-s + 3.71·29-s + 3.65·43-s − 2/7·49-s − 1.64·53-s − 0.512·61-s + 3/8·64-s + 1.45·68-s + 1.80·79-s + 9/5·100-s − 3.58·101-s − 2.32·107-s + 1.12·113-s + 5.57·116-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(147.507\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1521} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2313441,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.485991728\)
\(L(\frac12)\) \(\approx\) \(4.485991728\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
13 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 174 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−9.632329551395308856333565239859, −9.384373766189590463031197894900, −8.835438824409376606960205624597, −8.334248954155828532794708635046, −7.918147403743473727379022895859, −7.81781235133689436226841505768, −7.02005853911333667455607411882, −6.93849291761639245410735055019, −6.48650284417218830676544772663, −6.05535666758222217048917172906, −5.75401838951053768388123523620, −5.15244582722234115187458525947, −4.55083173295707621349601812299, −4.35761070343197201943379184790, −3.44424116905537115240067014073, −2.99835845276749615904355331952, −2.66683722585971036089659170777, −2.24083217726722598682601865848, −1.21592559322955815662451966162, −0.986715544991487796685379614942, 0.986715544991487796685379614942, 1.21592559322955815662451966162, 2.24083217726722598682601865848, 2.66683722585971036089659170777, 2.99835845276749615904355331952, 3.44424116905537115240067014073, 4.35761070343197201943379184790, 4.55083173295707621349601812299, 5.15244582722234115187458525947, 5.75401838951053768388123523620, 6.05535666758222217048917172906, 6.48650284417218830676544772663, 6.93849291761639245410735055019, 7.02005853911333667455607411882, 7.81781235133689436226841505768, 7.918147403743473727379022895859, 8.334248954155828532794708635046, 8.835438824409376606960205624597, 9.384373766189590463031197894900, 9.632329551395308856333565239859

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