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Properties

Label 4-50e2-1.1-c25e2-0-2
Degree $4$
Conductor $2500$
Sign $1$
Analytic cond. $39203.3$
Root an. cond. $14.0711$
Motivic weight $25$
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  − 1.67e7·4-s + 1.68e12·9-s − 2.90e13·11-s + 2.81e14·16-s + 2.79e16·19-s − 4.16e18·29-s + 5.32e18·31-s − 2.82e19·36-s + 4.67e20·41-s + 4.86e20·44-s + 1.01e21·49-s + 1.66e22·59-s + 4.85e22·61-s − 4.72e21·64-s − 1.86e23·71-s − 4.69e23·76-s + 1.61e24·79-s + 2.12e24·81-s − 7.11e24·89-s − 4.88e25·99-s + 1.45e25·101-s + 3.32e25·109-s + 6.98e25·116-s + 4.14e26·121-s − 8.93e25·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.98·9-s − 2.78·11-s + 1/4·16-s + 2.90·19-s − 2.18·29-s + 1.21·31-s − 0.994·36-s + 3.23·41-s + 1.39·44-s + 0.753·49-s + 1.22·59-s + 2.34·61-s − 1/8·64-s − 1.34·71-s − 1.45·76-s + 3.06·79-s + 2.95·81-s − 3.05·89-s − 5.54·99-s + 1.28·101-s + 1.13·109-s + 1.09·116-s + 3.82·121-s − 0.607·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(39203.3\)
Root analytic conductor: \(14.0711\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2500,\ (\ :25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(3.590055419\)
\(L(\frac12)\) \(\approx\) \(3.590055419\)
\(L(\frac{27}{2})\) 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$C_2$ \( 1 + p^{24} T^{2} \)
5 \( 1 \)
good3$C_2^2$ \( 1 - 2311360550 p^{6} T^{2} + p^{50} T^{4} \)
7$C_2^2$ \( 1 - \)\(10\!\cdots\!90\)\( T^{2} + p^{50} T^{4} \)
11$C_2$ \( ( 1 + 119886135348 p^{2} T + p^{25} T^{2} )^{2} \)
13$C_2^2$ \( 1 - \)\(63\!\cdots\!50\)\( T^{2} + p^{50} T^{4} \)
17$C_2^2$ \( 1 - \)\(15\!\cdots\!30\)\( p^{2} T^{2} + p^{50} T^{4} \)
19$C_2$ \( ( 1 - 736811826531460 p T + p^{25} T^{2} )^{2} \)
23$C_2^2$ \( 1 - \)\(27\!\cdots\!70\)\( p^{2} T^{2} + p^{50} T^{4} \)
29$C_2$ \( ( 1 + 2080230429601526910 T + p^{25} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2663532371302675232 T + p^{25} T^{2} )^{2} \)
37$C_2^2$ \( 1 - \)\(56\!\cdots\!90\)\( T^{2} + p^{50} T^{4} \)
41$C_2$ \( ( 1 - \)\(23\!\cdots\!22\)\( T + p^{25} T^{2} )^{2} \)
43$C_2^2$ \( 1 - \)\(13\!\cdots\!30\)\( T^{2} + p^{50} T^{4} \)
47$C_2^2$ \( 1 - \)\(11\!\cdots\!30\)\( T^{2} + p^{50} T^{4} \)
53$C_2^2$ \( 1 - \)\(25\!\cdots\!10\)\( T^{2} + p^{50} T^{4} \)
59$C_2$ \( ( 1 - \)\(83\!\cdots\!80\)\( T + p^{25} T^{2} )^{2} \)
61$C_2$ \( ( 1 - \)\(24\!\cdots\!62\)\( T + p^{25} T^{2} )^{2} \)
67$C_2^2$ \( 1 + \)\(65\!\cdots\!70\)\( T^{2} + p^{50} T^{4} \)
71$C_2$ \( ( 1 + \)\(93\!\cdots\!88\)\( T + p^{25} T^{2} )^{2} \)
73$C_2^2$ \( 1 - \)\(74\!\cdots\!90\)\( T^{2} + p^{50} T^{4} \)
79$C_2$ \( ( 1 - \)\(80\!\cdots\!80\)\( T + p^{25} T^{2} )^{2} \)
83$C_2^2$ \( 1 - \)\(18\!\cdots\!10\)\( T^{2} + p^{50} T^{4} \)
89$C_2$ \( ( 1 + \)\(35\!\cdots\!90\)\( T + p^{25} T^{2} )^{2} \)
97$C_2^2$ \( 1 - \)\(18\!\cdots\!90\)\( T^{2} + p^{50} T^{4} \)
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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

−11.56910619768844341456790390458, −10.32247826475391347518613669076, −10.10886845509996706379191077376, −9.539517750479977680472315560784, −9.241095860485609669822128988065, −8.074120073125140872709118209134, −7.80631201524660919398038165262, −7.29687387727108583570474859522, −7.14274003614780968625282130884, −5.84700779839146239820219516712, −5.47219486260854562763706443049, −5.05635649389781934395349442503, −4.50878727406498061928764319524, −3.83540640228395727319647498258, −3.34887640491460949794846702058, −2.43454575719537163880212140940, −2.34856068818202039546226208422, −1.25973068833346783059988508179, −0.900776489494010525656916243899, −0.43595442275936098591851901586, 0.43595442275936098591851901586, 0.900776489494010525656916243899, 1.25973068833346783059988508179, 2.34856068818202039546226208422, 2.43454575719537163880212140940, 3.34887640491460949794846702058, 3.83540640228395727319647498258, 4.50878727406498061928764319524, 5.05635649389781934395349442503, 5.47219486260854562763706443049, 5.84700779839146239820219516712, 7.14274003614780968625282130884, 7.29687387727108583570474859522, 7.80631201524660919398038165262, 8.074120073125140872709118209134, 9.241095860485609669822128988065, 9.539517750479977680472315560784, 10.10886845509996706379191077376, 10.32247826475391347518613669076, 11.56910619768844341456790390458

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