Properties

Label 4-10e4-1.1-c8e2-0-1
Degree $4$
Conductor $10000$
Sign $1$
Analytic cond. $1659.57$
Root an. cond. $6.38262$
Motivic weight $8$
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  + 20·2-s + 144·4-s − 2.24e3·8-s + 3.13e3·9-s + 1.09e4·13-s − 8.16e4·16-s − 1.46e5·17-s + 6.27e4·18-s + 2.18e5·26-s − 2.56e5·29-s − 1.05e6·32-s − 2.92e6·34-s + 4.51e5·36-s + 6.94e6·37-s + 4.29e6·41-s + 9.57e6·49-s + 1.57e6·52-s − 1.64e6·53-s − 5.12e6·58-s − 2.94e7·61-s − 2.90e5·64-s − 2.10e7·68-s − 7.02e6·72-s + 1.14e7·73-s + 1.38e8·74-s − 3.31e7·81-s + 8.58e7·82-s + ⋯
L(s)  = 1  + 5/4·2-s + 9/16·4-s − 0.546·8-s + 0.478·9-s + 0.383·13-s − 1.24·16-s − 1.75·17-s + 0.597·18-s + 0.478·26-s − 0.362·29-s − 1.01·32-s − 2.18·34-s + 0.269·36-s + 3.70·37-s + 1.51·41-s + 1.66·49-s + 0.215·52-s − 0.208·53-s − 0.453·58-s − 2.13·61-s − 0.0173·64-s − 0.984·68-s − 0.261·72-s + 0.403·73-s + 4.63·74-s − 0.771·81-s + 1.89·82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10000\)    =    \(2^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1659.57\)
Root analytic conductor: \(6.38262\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10000,\ (\ :4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.821758168\)
\(L(\frac12)\) \(\approx\) \(3.821758168\)
\(L(5)\) 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$C_2$ \( 1 - 5 p^{2} T + p^{8} T^{2} \)
5 \( 1 \)
good3$C_2^2$ \( 1 - 1046 p T^{2} + p^{16} T^{4} \)
7$C_2^2$ \( 1 - 195362 p^{2} T^{2} + p^{16} T^{4} \)
11$C_2^2$ \( 1 - 87015362 T^{2} + p^{16} T^{4} \)
13$C_2$ \( ( 1 - 5470 T + p^{8} T^{2} )^{2} \)
17$C_2$ \( ( 1 + 73090 T + p^{8} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 33587484482 T^{2} + p^{16} T^{4} \)
23$C_2^2$ \( 1 - 100353384578 T^{2} + p^{16} T^{4} \)
29$C_2$ \( ( 1 + 128222 T + p^{8} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 1701165473282 T^{2} + p^{16} T^{4} \)
37$C_2$ \( ( 1 - 3472030 T + p^{8} T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2146882 T + p^{8} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 11766582970942 T^{2} + p^{16} T^{4} \)
47$C_2^2$ \( 1 + 10537750788862 T^{2} + p^{16} T^{4} \)
53$C_2$ \( ( 1 + 824290 T + p^{8} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 279781405698242 T^{2} + p^{16} T^{4} \)
61$C_2$ \( ( 1 + 14746078 T + p^{8} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 579369070794818 T^{2} + p^{16} T^{4} \)
71$C_2^2$ \( 1 - 1290076545985922 T^{2} + p^{16} T^{4} \)
73$C_2$ \( ( 1 - 5725630 T + p^{8} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 1744457179595522 T^{2} + p^{16} T^{4} \)
83$C_2^2$ \( 1 - 1804713576833858 T^{2} + p^{16} T^{4} \)
89$C_2$ \( ( 1 + 83324222 T + p^{8} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 120619010 T + p^{8} T^{2} )^{2} \)
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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

−13.06274448365495508289800426144, −12.16523050872682017951143637997, −11.54449094768855506138710161045, −10.91487486490400093165662042445, −10.88406627294641026297676485328, −9.597106663453854062282133004488, −9.457158896145852434135632415622, −8.769274987907796786415304895002, −8.079854756701911375413140979822, −7.35401758166011095200879046607, −6.74764006402317521654792801658, −6.02763873878261766002209968218, −5.79585884636010310450055239232, −4.76364072246339087921857912606, −4.14735060784961943484962439741, −4.09831564197085165340472669269, −2.75124450073670663198300714436, −2.52555440435739325971294378086, −1.39123682683298762205799344567, −0.45430277484204201231094552961, 0.45430277484204201231094552961, 1.39123682683298762205799344567, 2.52555440435739325971294378086, 2.75124450073670663198300714436, 4.09831564197085165340472669269, 4.14735060784961943484962439741, 4.76364072246339087921857912606, 5.79585884636010310450055239232, 6.02763873878261766002209968218, 6.74764006402317521654792801658, 7.35401758166011095200879046607, 8.079854756701911375413140979822, 8.769274987907796786415304895002, 9.457158896145852434135632415622, 9.597106663453854062282133004488, 10.88406627294641026297676485328, 10.91487486490400093165662042445, 11.54449094768855506138710161045, 12.16523050872682017951143637997, 13.06274448365495508289800426144

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