Properties

Label 4-10e4-1.1-c3e2-0-2
Degree $4$
Conductor $10000$
Sign $1$
Analytic cond. $34.8122$
Root an. cond. $2.42903$
Motivic weight $3$
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  + 38·9-s − 120·11-s − 88·19-s + 372·29-s + 352·31-s + 372·41-s + 430·49-s + 504·59-s − 116·61-s + 336·71-s − 544·79-s + 715·81-s + 2.02e3·89-s − 4.56e3·99-s − 2.62e3·101-s − 556·109-s + 8.13e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3.00e3·169-s + ⋯
L(s)  = 1  + 1.40·9-s − 3.28·11-s − 1.06·19-s + 2.38·29-s + 2.03·31-s + 1.41·41-s + 1.25·49-s + 1.11·59-s − 0.243·61-s + 0.561·71-s − 0.774·79-s + 0.980·81-s + 2.41·89-s − 4.62·99-s − 2.58·101-s − 0.488·109-s + 6.11·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.36·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10000\)    =    \(2^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(34.8122\)
Root analytic conductor: \(2.42903\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10000,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.645880601\)
\(L(\frac12)\) \(\approx\) \(1.645880601\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 - 38 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 - 430 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 60 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 3002 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 9502 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 + 44 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 22030 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 186 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 176 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 36790 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 186 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 149014 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 179422 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 49750 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 252 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 58 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 471770 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 168 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 521998 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 + 272 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 244870 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 1014 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 1238590 T^{2} + p^{6} T^{4} \)
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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

−13.76492553605150408944691994168, −13.05981286787324841657774721561, −12.64264936783253109661796366932, −12.36347796463015559510327025192, −11.54868313645862562245696587039, −10.58094451730751497712635795310, −10.42062042623944260433717825104, −10.24567351412919993094495006350, −9.478506394283487437644763242830, −8.427876360574542047146671837201, −8.139596549323974062959064948580, −7.62191911276852733035021536432, −6.92854312542438817913821164950, −6.24818868956177893080051739332, −5.37652633816202316078484930566, −4.73317541277584288923263248815, −4.26474664466351236765663714865, −2.79897712720710405170732166234, −2.39419980352675496741695973694, −0.73684304630943648114370250091, 0.73684304630943648114370250091, 2.39419980352675496741695973694, 2.79897712720710405170732166234, 4.26474664466351236765663714865, 4.73317541277584288923263248815, 5.37652633816202316078484930566, 6.24818868956177893080051739332, 6.92854312542438817913821164950, 7.62191911276852733035021536432, 8.139596549323974062959064948580, 8.427876360574542047146671837201, 9.478506394283487437644763242830, 10.24567351412919993094495006350, 10.42062042623944260433717825104, 10.58094451730751497712635795310, 11.54868313645862562245696587039, 12.36347796463015559510327025192, 12.64264936783253109661796366932, 13.05981286787324841657774721561, 13.76492553605150408944691994168

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