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Properties

Label 4-80e4-1.1-c1e2-0-4
Degree $4$
Conductor $40960000$
Sign $1$
Analytic cond. $2611.64$
Root an. cond. $7.14872$
Motivic weight $1$
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  − 6·9-s + 4·11-s − 12·19-s − 4·41-s + 6·49-s + 28·59-s + 27·81-s − 28·89-s − 24·99-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·169-s + 72·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 2·9-s + 1.20·11-s − 2.75·19-s − 0.624·41-s + 6/7·49-s + 3.64·59-s + 3·81-s − 2.96·89-s − 2.41·99-s − 0.909·121-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 − 0.461·169-s + 5.50·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40960000\)    =    \(2^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2611.64\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 40960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.472985304\)
\(L(\frac12)\) \(\approx\) \(1.472985304\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
5 \( 1 \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.19.m_cw
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.23.a_ba
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.37.a_cc
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.47.a_adi
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.53.a_acw
59$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.59.abc_mc
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.73.a_fq
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.89.bc_ok
97$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.97.a_hm
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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

−8.315712647209521079979155237226, −8.157596144591216785221095324323, −7.43856900649409982691375733078, −7.01200744609270274511866773048, −6.69934691780867890438032075559, −6.52104739049924581790185829593, −6.03695639274537178144390386610, −5.81869608323955159768823193478, −5.30717235200283562553062365044, −5.22951401649411886111213490444, −4.39275711684487283759729653047, −4.17022715486239219539841764160, −3.95877098433259406134766107408, −3.38297719259055656703215775198, −2.93291257552142255788145678193, −2.56389319830202201244705472492, −1.99581255409106659803472404888, −1.85309766336570377699208936780, −0.850970556049532310052870196815, −0.37017107729235447756697027853, 0.37017107729235447756697027853, 0.850970556049532310052870196815, 1.85309766336570377699208936780, 1.99581255409106659803472404888, 2.56389319830202201244705472492, 2.93291257552142255788145678193, 3.38297719259055656703215775198, 3.95877098433259406134766107408, 4.17022715486239219539841764160, 4.39275711684487283759729653047, 5.22951401649411886111213490444, 5.30717235200283562553062365044, 5.81869608323955159768823193478, 6.03695639274537178144390386610, 6.52104739049924581790185829593, 6.69934691780867890438032075559, 7.01200744609270274511866773048, 7.43856900649409982691375733078, 8.157596144591216785221095324323, 8.315712647209521079979155237226

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