Properties

Label 4-1568e2-1.1-c2e2-0-0
Degree $4$
Conductor $2458624$
Sign $1$
Analytic cond. $1825.41$
Root an. cond. $6.53642$
Motivic weight $2$
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  − 2·5-s + 17·9-s − 48·13-s − 2·17-s − 47·25-s + 48·29-s − 98·37-s + 96·41-s − 34·45-s − 50·53-s + 2·61-s + 96·65-s − 190·73-s + 208·81-s + 4·85-s − 190·89-s − 288·97-s − 146·101-s − 142·109-s + 192·113-s − 816·117-s − 47·121-s + 146·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 2/5·5-s + 17/9·9-s − 3.69·13-s − 0.117·17-s − 1.87·25-s + 1.65·29-s − 2.64·37-s + 2.34·41-s − 0.755·45-s − 0.943·53-s + 2/61·61-s + 1.47·65-s − 2.60·73-s + 2.56·81-s + 4/85·85-s − 2.13·89-s − 2.96·97-s − 1.44·101-s − 1.30·109-s + 1.69·113-s − 6.97·117-s − 0.388·121-s + 1.16·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2458624\)    =    \(2^{10} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1825.41\)
Root analytic conductor: \(6.53642\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2458624,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.02037500047\)
\(L(\frac12)\) \(\approx\) \(0.02037500047\)
\(L(2)\) 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 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 17 T^{2} + p^{4} T^{4} \)
5$C_2$ \( ( 1 + T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 + 47 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 24 T + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 + T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 673 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - 1009 T^{2} + p^{4} T^{4} \)
29$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 241 T^{2} + p^{4} T^{4} \)
37$C_2$ \( ( 1 + 49 T + p^{2} T^{2} )^{2} \)
41$C_2$ \( ( 1 - 48 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 3122 T^{2} + p^{4} T^{4} \)
47$C_2^2$ \( 1 - 1393 T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 + 25 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 6673 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 4753 T^{2} + p^{4} T^{4} \)
71$C_2^2$ \( 1 - 866 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 95 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 10801 T^{2} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 8594 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 + 95 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 144 T + p^{2} T^{2} )^{2} \)
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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

−9.585671589860434202168201371447, −9.208845359957239238337697286015, −8.684249048673170416666209227502, −7.973158524482111106585256454299, −7.82994512642282414238231101193, −7.26532360147701610936480437099, −7.24493519917293213811890555274, −6.83378794036345885091336655283, −6.39024359585340319699812337716, −5.44964170641966762550519063799, −5.44252710686740174706948520646, −4.56957747946361817481303712848, −4.55318899343814814947202738878, −4.18926273854097998308499131918, −3.53569745841225698389206515271, −2.67428628889902789518317161807, −2.52668993365403805641694438733, −1.79748825101306848895504491385, −1.26876705039339875343631158995, −0.03867552626891854107382691626, 0.03867552626891854107382691626, 1.26876705039339875343631158995, 1.79748825101306848895504491385, 2.52668993365403805641694438733, 2.67428628889902789518317161807, 3.53569745841225698389206515271, 4.18926273854097998308499131918, 4.55318899343814814947202738878, 4.56957747946361817481303712848, 5.44252710686740174706948520646, 5.44964170641966762550519063799, 6.39024359585340319699812337716, 6.83378794036345885091336655283, 7.24493519917293213811890555274, 7.26532360147701610936480437099, 7.82994512642282414238231101193, 7.973158524482111106585256454299, 8.684249048673170416666209227502, 9.208845359957239238337697286015, 9.585671589860434202168201371447

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