Properties

Label 4-224e2-1.1-c5e2-0-2
Degree $4$
Conductor $50176$
Sign $1$
Analytic cond. $1290.67$
Root an. cond. $5.99382$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 14·3-s + 34·5-s + 98·7-s − 278·9-s − 420·11-s − 490·13-s + 476·15-s − 1.05e3·17-s − 1.24e3·19-s + 1.37e3·21-s + 504·23-s − 2.39e3·25-s − 7.12e3·27-s − 3.90e3·29-s − 2.04e3·31-s − 5.88e3·33-s + 3.33e3·35-s − 7.48e3·37-s − 6.86e3·39-s + 7.83e3·41-s − 1.03e4·43-s − 9.45e3·45-s − 4.19e4·47-s + 7.20e3·49-s − 1.47e4·51-s + 3.28e4·53-s − 1.42e4·55-s + ⋯
L(s)  = 1  + 0.898·3-s + 0.608·5-s + 0.755·7-s − 1.14·9-s − 1.04·11-s − 0.804·13-s + 0.546·15-s − 0.886·17-s − 0.791·19-s + 0.678·21-s + 0.198·23-s − 0.766·25-s − 1.88·27-s − 0.862·29-s − 0.382·31-s − 0.939·33-s + 0.459·35-s − 0.899·37-s − 0.722·39-s + 0.727·41-s − 0.852·43-s − 0.695·45-s − 2.77·47-s + 3/7·49-s − 0.795·51-s + 1.60·53-s − 0.636·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50176\)    =    \(2^{10} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1290.67\)
Root analytic conductor: \(5.99382\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 50176,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 - p^{2} T )^{2} \)
good3$D_{4}$ \( 1 - 14 T + 158 p T^{2} - 14 p^{5} T^{3} + p^{10} T^{4} \)
5$D_{4}$ \( 1 - 34 T + 142 p^{2} T^{2} - 34 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 + 420 T + 237126 T^{2} + 420 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 490 T + 656150 T^{2} + 490 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 1056 T + 3106542 T^{2} + 1056 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 1246 T + 1618778 T^{2} + 1246 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 504 T + 12920574 T^{2} - 504 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 3904 T + 26647526 T^{2} + 3904 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 2044 T + 33160782 T^{2} + 2044 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 7488 T + 152693494 T^{2} + 7488 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 7832 T + 168604142 T^{2} - 7832 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 10332 T + 248230342 T^{2} + 10332 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 41972 T + 855029710 T^{2} + 41972 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 32812 T + 664801838 T^{2} - 32812 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 48398 T + 1851574618 T^{2} + 48398 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 718 T + 1677458142 T^{2} + 718 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 12824 T + 1908078582 T^{2} + 12824 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 103992 T + 6302476942 T^{2} + 103992 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 54100 T + 3096449510 T^{2} + 54100 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 64568 T + 7121481950 T^{2} + 64568 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 47810 T + 8432588842 T^{2} - 47810 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 17388 T - 1489722410 T^{2} + 17388 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 97296 T + 17237136142 T^{2} + 97296 p^{5} T^{3} + p^{10} T^{4} \)
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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

−11.07647855269186025809172753379, −10.68314610787122361674504227114, −10.14462441451551085826847267496, −9.628360931402863298881169495335, −8.938857550567959566632331451841, −8.812306754265610636529917824803, −8.143664768615809890241917948099, −7.83388689719900286671753884757, −7.25856585206097244256144596072, −6.56583246339130667938672495269, −5.69796808371529532917433742395, −5.58783773619222350943544900848, −4.77377101626411512920540647055, −4.25415424526068915276777418624, −3.17848997687018542076802660632, −2.84943190522414720071472136886, −1.91101720690821588279249872130, −1.89248897361685020734015824899, 0, 0, 1.89248897361685020734015824899, 1.91101720690821588279249872130, 2.84943190522414720071472136886, 3.17848997687018542076802660632, 4.25415424526068915276777418624, 4.77377101626411512920540647055, 5.58783773619222350943544900848, 5.69796808371529532917433742395, 6.56583246339130667938672495269, 7.25856585206097244256144596072, 7.83388689719900286671753884757, 8.143664768615809890241917948099, 8.812306754265610636529917824803, 8.938857550567959566632331451841, 9.628360931402863298881169495335, 10.14462441451551085826847267496, 10.68314610787122361674504227114, 11.07647855269186025809172753379

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