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Properties

Label 4-416000-1.1-c1e2-0-5
Degree $4$
Conductor $416000$
Sign $1$
Analytic cond. $26.5245$
Root an. cond. $2.26940$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 3·9-s + 15-s + 25-s − 4·27-s − 5·31-s + 14·37-s − 3·41-s + 8·43-s − 3·45-s + 5·49-s − 9·53-s + 8·67-s + 3·71-s + 75-s + 25·79-s + 2·81-s + 3·89-s − 5·93-s + 14·111-s + 20·121-s − 3·123-s + 125-s + 127-s + 8·129-s + 131-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 9-s + 0.258·15-s + 1/5·25-s − 0.769·27-s − 0.898·31-s + 2.30·37-s − 0.468·41-s + 1.21·43-s − 0.447·45-s + 5/7·49-s − 1.23·53-s + 0.977·67-s + 0.356·71-s + 0.115·75-s + 2.81·79-s + 2/9·81-s + 0.317·89-s − 0.518·93-s + 1.32·111-s + 1.81·121-s − 0.270·123-s + 0.0894·125-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(416000\)    =    \(2^{8} \cdot 5^{3} \cdot 13\)
Sign: $1$
Analytic conductor: \(26.5245\)
Root analytic conductor: \(2.26940\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 416000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.141228278\)
\(L(\frac12)\) \(\approx\) \(2.141228278\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - T + p T^{2} ) \)
good3$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.ab_e
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.11.a_au
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.19.a_bc
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.29.a_bi
31$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.f_co
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.37.ao_ek
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.41.d_de
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \) 2.47.a_bl
53$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.j_cs
59$C_2^2$ \( 1 - 116 T^{2} + p^{2} T^{4} \) 2.59.a_aem
61$C_2^2$ \( 1 - 89 T^{2} + p^{2} T^{4} \) 2.61.a_adl
67$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.ai_cr
71$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.ad_abm
73$C_2^2$ \( 1 - 71 T^{2} + p^{2} T^{4} \) 2.73.a_act
79$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.79.az_li
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.ad_cs
97$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.97.a_t
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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

−8.744366244332160258213147604362, −8.136130804916435083548943140734, −7.76513424888821708258199154211, −7.46138896248954653094546129348, −6.68090479383506612086531062741, −6.26514125671461671021940770726, −5.87762842983473907520289444941, −5.33178396974182342337405072549, −4.88065411300981315104598920779, −4.15926715015460879204798328299, −3.63238644278741619632373268737, −2.98212283208671920725303181409, −2.48911703396987557234828169690, −1.92479522432444201826489410957, −0.77088117838219230527668027430, 0.77088117838219230527668027430, 1.92479522432444201826489410957, 2.48911703396987557234828169690, 2.98212283208671920725303181409, 3.63238644278741619632373268737, 4.15926715015460879204798328299, 4.88065411300981315104598920779, 5.33178396974182342337405072549, 5.87762842983473907520289444941, 6.26514125671461671021940770726, 6.68090479383506612086531062741, 7.46138896248954653094546129348, 7.76513424888821708258199154211, 8.136130804916435083548943140734, 8.744366244332160258213147604362

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