Properties

Label 4-50e2-1.1-c21e2-0-1
Degree $4$
Conductor $2500$
Sign $1$
Analytic cond. $19526.8$
Root an. cond. $11.8211$
Motivic weight $21$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.04e6·4-s + 1.57e10·9-s + 1.73e11·11-s + 1.09e12·16-s − 4.60e13·19-s + 1.46e15·29-s − 6.29e15·31-s − 1.65e16·36-s + 9.14e16·41-s − 1.81e17·44-s + 3.89e17·49-s + 7.56e18·59-s − 1.52e19·61-s − 1.15e18·64-s − 9.05e18·71-s + 4.83e19·76-s − 1.98e20·79-s + 1.40e20·81-s − 2.37e20·89-s + 2.73e21·99-s − 5.30e20·101-s − 6.67e21·109-s − 1.53e21·116-s + 7.76e21·121-s + 6.59e21·124-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.50·9-s + 2.01·11-s + 1/4·16-s − 1.72·19-s + 0.648·29-s − 1.37·31-s − 0.754·36-s + 1.06·41-s − 1.00·44-s + 0.696·49-s + 1.92·59-s − 2.73·61-s − 1/8·64-s − 0.330·71-s + 0.861·76-s − 2.36·79-s + 1.27·81-s − 0.807·89-s + 3.04·99-s − 0.477·101-s − 2.70·109-s − 0.324·116-s + 1.04·121-s + 0.689·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(19526.8\)
Root analytic conductor: \(11.8211\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2500,\ (\ :21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(2.197129919\)
\(L(\frac12)\) \(\approx\) \(2.197129919\)
\(L(\frac{23}{2})\) 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$C_2$ \( 1 + p^{20} T^{2} \)
5 \( 1 \)
good3$C_2^2$ \( 1 - 2407190 p^{8} T^{2} + p^{42} T^{4} \)
7$C_2^2$ \( 1 - 7941579723511150 p^{2} T^{2} + p^{42} T^{4} \)
11$C_2$ \( ( 1 - 7884652692 p T + p^{21} T^{2} )^{2} \)
13$C_2^2$ \( 1 + \)\(30\!\cdots\!70\)\( T^{2} + p^{42} T^{4} \)
17$C_2^2$ \( 1 - \)\(44\!\cdots\!90\)\( p^{2} T^{2} + p^{42} T^{4} \)
19$C_2$ \( ( 1 + 1212235139180 p T + p^{21} T^{2} )^{2} \)
23$C_2^2$ \( 1 - \)\(57\!\cdots\!70\)\( T^{2} + p^{42} T^{4} \)
29$C_2$ \( ( 1 - 734051633521170 T + p^{21} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 3146664162057568 T + p^{21} T^{2} )^{2} \)
37$C_2^2$ \( 1 - \)\(15\!\cdots\!30\)\( T^{2} + p^{42} T^{4} \)
41$C_2$ \( ( 1 - 45714648841476042 T + p^{21} T^{2} )^{2} \)
43$C_2^2$ \( 1 - \)\(39\!\cdots\!50\)\( T^{2} + p^{42} T^{4} \)
47$C_2^2$ \( 1 - \)\(57\!\cdots\!90\)\( T^{2} + p^{42} T^{4} \)
53$C_2^2$ \( 1 + \)\(10\!\cdots\!10\)\( T^{2} + p^{42} T^{4} \)
59$C_2$ \( ( 1 - 3780497099978396340 T + p^{21} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 7619813346829729138 T + p^{21} T^{2} )^{2} \)
67$C_2^2$ \( 1 - \)\(92\!\cdots\!10\)\( T^{2} + p^{42} T^{4} \)
71$C_2$ \( ( 1 + 4526486567453771928 T + p^{21} T^{2} )^{2} \)
73$C_2^2$ \( 1 - \)\(20\!\cdots\!70\)\( T^{2} + p^{42} T^{4} \)
79$C_2$ \( ( 1 + 99336442530925070480 T + p^{21} T^{2} )^{2} \)
83$C_2^2$ \( 1 - \)\(39\!\cdots\!10\)\( T^{2} + p^{42} T^{4} \)
89$C_2$ \( ( 1 + \)\(11\!\cdots\!90\)\( T + p^{21} T^{2} )^{2} \)
97$C_2^2$ \( 1 - \)\(73\!\cdots\!90\)\( T^{2} + p^{42} T^{4} \)
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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

−12.17353245203254450024984872312, −10.86711124740368675642674380152, −10.70206063683238196835383542975, −9.753766483359357318616796002707, −9.564958831726811156665950002592, −8.745507369355611425893436275733, −8.607713382864770684644181936703, −7.51898062573216813335713698643, −7.14237640540903218473166381959, −6.47733256628916341646336513870, −6.11879974996111301397018440080, −5.24290729183518708497856882071, −4.38853788091406564196720895965, −4.08442027957201575427829574279, −3.88761926788114734625837394582, −2.86879987016955764802847820089, −2.04072672457927557809682042989, −1.38824926949537898844379134907, −1.20447549963689638552495716423, −0.31068542135626789157765065673, 0.31068542135626789157765065673, 1.20447549963689638552495716423, 1.38824926949537898844379134907, 2.04072672457927557809682042989, 2.86879987016955764802847820089, 3.88761926788114734625837394582, 4.08442027957201575427829574279, 4.38853788091406564196720895965, 5.24290729183518708497856882071, 6.11879974996111301397018440080, 6.47733256628916341646336513870, 7.14237640540903218473166381959, 7.51898062573216813335713698643, 8.607713382864770684644181936703, 8.745507369355611425893436275733, 9.564958831726811156665950002592, 9.753766483359357318616796002707, 10.70206063683238196835383542975, 10.86711124740368675642674380152, 12.17353245203254450024984872312

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