Properties

Label 4-68e2-1.1-c4e2-0-0
Degree $4$
Conductor $4624$
Sign $1$
Analytic cond. $49.4090$
Root an. cond. $2.65125$
Motivic weight $4$
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  − 16·4-s − 34·5-s − 480·13-s + 256·16-s − 322·17-s + 544·20-s + 578·25-s − 1.59e3·29-s − 3.84e3·37-s − 1.59e3·41-s + 7.68e3·52-s + 4.31e3·61-s − 4.09e3·64-s + 1.63e4·65-s + 5.15e3·68-s + 1.20e4·73-s − 8.70e3·80-s − 6.56e3·81-s + 1.09e4·85-s − 2.49e4·89-s + 2.06e4·97-s − 9.24e3·100-s + 1.58e4·101-s + 1.24e4·109-s + 1.79e4·113-s + 2.55e4·116-s − 2.12e4·125-s + ⋯
L(s)  = 1  − 4-s − 1.35·5-s − 2.84·13-s + 16-s − 1.11·17-s + 1.35·20-s + 0.924·25-s − 1.90·29-s − 2.80·37-s − 0.950·41-s + 2.84·52-s + 1.16·61-s − 64-s + 3.86·65-s + 1.11·68-s + 2.25·73-s − 1.35·80-s − 81-s + 1.51·85-s − 3.15·89-s + 2.19·97-s − 0.924·100-s + 1.55·101-s + 1.05·109-s + 1.40·113-s + 1.90·116-s − 1.35·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4624\)    =    \(2^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(49.4090\)
Root analytic conductor: \(2.65125\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4624,\ (\ :2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0005194433078\)
\(L(\frac12)\) \(\approx\) \(0.0005194433078\)
\(L(3)\) 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$C_2$ \( 1 + p^{4} T^{2} \)
17$C_2$ \( 1 + 322 T + p^{4} T^{2} \)
good3$C_2^2$ \( 1 + p^{8} T^{4} \)
5$C_2$ \( ( 1 - 14 T + p^{4} T^{2} )( 1 + 48 T + p^{4} T^{2} ) \)
7$C_2^2$ \( 1 + p^{8} T^{4} \)
11$C_2^2$ \( 1 + p^{8} T^{4} \)
13$C_2$ \( ( 1 + 240 T + p^{4} T^{2} )^{2} \)
19$C_2$ \( ( 1 + p^{4} T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{8} T^{4} \)
29$C_2$ \( ( 1 - 82 T + p^{4} T^{2} )( 1 + 1680 T + p^{4} T^{2} ) \)
31$C_2^2$ \( 1 + p^{8} T^{4} \)
37$C_2$ \( ( 1 + 1680 T + p^{4} T^{2} )( 1 + 2162 T + p^{4} T^{2} ) \)
41$C_2$ \( ( 1 - 1440 T + p^{4} T^{2} )( 1 + 3038 T + p^{4} T^{2} ) \)
43$C_2$ \( ( 1 + p^{4} T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
53$C_2$ \( ( 1 - 2482 T + p^{4} T^{2} )( 1 + 2482 T + p^{4} T^{2} ) \)
59$C_2$ \( ( 1 + p^{4} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6958 T + p^{4} T^{2} )( 1 + 2640 T + p^{4} T^{2} ) \)
67$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
71$C_2^2$ \( 1 + p^{8} T^{4} \)
73$C_2$ \( ( 1 - 10560 T + p^{4} T^{2} )( 1 - 1442 T + p^{4} T^{2} ) \)
79$C_2^2$ \( 1 + p^{8} T^{4} \)
83$C_2$ \( ( 1 + p^{4} T^{2} )^{2} \)
89$C_2$ \( ( 1 + 12480 T + p^{4} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18720 T + p^{4} T^{2} )( 1 - 1918 T + p^{4} T^{2} ) \)
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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

−14.97152954424649128334358495527, −14.10419492835511078713885733152, −12.94726882385079371887987361012, −12.68239077092362601332805594278, −12.17559914991501170005584992506, −11.61009834454955228349232683276, −11.08607081626559005411674773584, −9.996998860814354939321159463853, −9.987440899112732180904059164062, −8.908281549867121101570835669852, −8.680889549002433773658074919595, −7.64740205512944680683115550140, −7.41577496693354809507373937242, −6.72843344538868948787304980904, −5.21830414548670572102105733607, −5.01603649752247585741162261375, −4.12681585531230022758155061165, −3.43980326353909428805831619891, −2.15686454539369302130736498381, −0.01186975090612976958539187239, 0.01186975090612976958539187239, 2.15686454539369302130736498381, 3.43980326353909428805831619891, 4.12681585531230022758155061165, 5.01603649752247585741162261375, 5.21830414548670572102105733607, 6.72843344538868948787304980904, 7.41577496693354809507373937242, 7.64740205512944680683115550140, 8.680889549002433773658074919595, 8.908281549867121101570835669852, 9.987440899112732180904059164062, 9.996998860814354939321159463853, 11.08607081626559005411674773584, 11.61009834454955228349232683276, 12.17559914991501170005584992506, 12.68239077092362601332805594278, 12.94726882385079371887987361012, 14.10419492835511078713885733152, 14.97152954424649128334358495527

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