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Properties

Label 4-4600e2-1.1-c1e2-0-5
Degree $4$
Conductor $21160000$
Sign $1$
Analytic cond. $1349.17$
Root an. cond. $6.06062$
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  + 5·9-s + 8·19-s + 6·29-s − 2·31-s − 10·41-s + 10·49-s + 6·71-s − 8·79-s + 16·81-s + 28·89-s + 4·101-s − 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 40·171-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 5/3·9-s + 1.83·19-s + 1.11·29-s − 0.359·31-s − 1.56·41-s + 10/7·49-s + 0.712·71-s − 0.900·79-s + 16/9·81-s + 2.96·89-s + 0.398·101-s − 0.383·109-s − 2·121-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 + 1.92·169-s + 3.05·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21160000\)    =    \(2^{6} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1349.17\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21160000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.727990412\)
\(L(\frac12)\) \(\approx\) \(3.727990412\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.3.a_af
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.13.a_az
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.29.ag_cp
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.31.c_cl
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.41.k_ed
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.47.a_an
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.53.a_ady
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.71.ag_fv
73$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \) 2.73.a_adt
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.83.a_ady
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.89.abc_ok
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.97.a_c
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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

−8.650524203935103463082212286254, −7.928854697608569505925354632955, −7.72765576009076395669394797812, −7.47319435499693160416064138720, −7.04356867610614022688941648505, −6.79340137275650513220145755505, −6.39693964969214777244098780027, −6.03021444292939359049747667735, −5.45073658544471289738114492843, −5.05684481265554933910522755670, −4.94079577608740647491928818249, −4.38317171589450461420408720334, −3.98588794714837404924292127733, −3.47146228558020806825342370411, −3.35337560757341363456061478935, −2.55786010747979372161898466282, −2.24731930222627930377552506802, −1.42222034249926501806183042596, −1.27481293674721069818678556489, −0.58151288401438805424188346575, 0.58151288401438805424188346575, 1.27481293674721069818678556489, 1.42222034249926501806183042596, 2.24731930222627930377552506802, 2.55786010747979372161898466282, 3.35337560757341363456061478935, 3.47146228558020806825342370411, 3.98588794714837404924292127733, 4.38317171589450461420408720334, 4.94079577608740647491928818249, 5.05684481265554933910522755670, 5.45073658544471289738114492843, 6.03021444292939359049747667735, 6.39693964969214777244098780027, 6.79340137275650513220145755505, 7.04356867610614022688941648505, 7.47319435499693160416064138720, 7.72765576009076395669394797812, 7.928854697608569505925354632955, 8.650524203935103463082212286254

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