Properties

Label 4-22e2-1.1-c13e2-0-0
Degree $4$
Conductor $484$
Sign $1$
Analytic cond. $556.526$
Root an. cond. $4.85703$
Motivic weight $13$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 128·2-s + 1.62e3·3-s + 1.22e4·4-s + 7.66e3·5-s − 2.08e5·6-s + 6.37e5·7-s − 1.04e6·8-s + 7.90e5·9-s − 9.81e5·10-s + 3.54e6·11-s + 1.99e7·12-s − 3.78e6·13-s − 8.15e7·14-s + 1.24e7·15-s + 8.38e7·16-s + 3.71e7·17-s − 1.01e8·18-s − 1.02e8·19-s + 9.41e7·20-s + 1.03e9·21-s − 4.53e8·22-s + 1.35e9·23-s − 1.70e9·24-s − 1.11e9·25-s + 4.84e8·26-s + 8.62e8·27-s + 7.82e9·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.28·3-s + 3/2·4-s + 0.219·5-s − 1.82·6-s + 2.04·7-s − 1.41·8-s + 0.495·9-s − 0.310·10-s + 0.603·11-s + 1.93·12-s − 0.217·13-s − 2.89·14-s + 0.282·15-s + 5/4·16-s + 0.373·17-s − 0.700·18-s − 0.499·19-s + 0.329·20-s + 2.63·21-s − 0.852·22-s + 1.90·23-s − 1.82·24-s − 0.914·25-s + 0.307·26-s + 0.428·27-s + 3.06·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(556.526\)
Root analytic conductor: \(4.85703\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 484,\ (\ :13/2, 13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(4.116220184\)
\(L(\frac12)\) \(\approx\) \(4.116220184\)
\(L(\frac{15}{2})\) 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$C_1$ \( ( 1 + p^{6} T )^{2} \)
11$C_1$ \( ( 1 - p^{6} T )^{2} \)
good3$D_{4}$ \( 1 - 542 p T + 68657 p^{3} T^{2} - 542 p^{14} T^{3} + p^{26} T^{4} \)
5$D_{4}$ \( 1 - 7666 T + 47007571 p^{2} T^{2} - 7666 p^{13} T^{3} + p^{26} T^{4} \)
7$D_{4}$ \( 1 - 637048 T + 6020286906 p^{2} T^{2} - 637048 p^{13} T^{3} + p^{26} T^{4} \)
13$D_{4}$ \( 1 + 291436 p T + 594138601660538 T^{2} + 291436 p^{14} T^{3} + p^{26} T^{4} \)
17$D_{4}$ \( 1 - 37137304 T + 19126613212775662 T^{2} - 37137304 p^{13} T^{3} + p^{26} T^{4} \)
19$D_{4}$ \( 1 + 102460596 T + 75240624868816198 T^{2} + 102460596 p^{13} T^{3} + p^{26} T^{4} \)
23$D_{4}$ \( 1 - 1352747042 T + 1373469205060873963 T^{2} - 1352747042 p^{13} T^{3} + p^{26} T^{4} \)
29$D_{4}$ \( 1 - 7425318120 T + 29655701354382887578 T^{2} - 7425318120 p^{13} T^{3} + p^{26} T^{4} \)
31$D_{4}$ \( 1 - 8163482594 T + 57290834624075598491 T^{2} - 8163482594 p^{13} T^{3} + p^{26} T^{4} \)
37$D_{4}$ \( 1 + 12073195594 T + 98252642021063024003 T^{2} + 12073195594 p^{13} T^{3} + p^{26} T^{4} \)
41$D_{4}$ \( 1 - 73792259580 T + \)\(31\!\cdots\!18\)\( T^{2} - 73792259580 p^{13} T^{3} + p^{26} T^{4} \)
43$D_{4}$ \( 1 - 20450919684 T + \)\(12\!\cdots\!46\)\( T^{2} - 20450919684 p^{13} T^{3} + p^{26} T^{4} \)
47$D_{4}$ \( 1 - 71306154600 T + \)\(56\!\cdots\!50\)\( T^{2} - 71306154600 p^{13} T^{3} + p^{26} T^{4} \)
53$D_{4}$ \( 1 - 309577967404 T + \)\(63\!\cdots\!46\)\( T^{2} - 309577967404 p^{13} T^{3} + p^{26} T^{4} \)
59$D_{4}$ \( 1 + 403802069082 T + \)\(23\!\cdots\!43\)\( T^{2} + 403802069082 p^{13} T^{3} + p^{26} T^{4} \)
61$D_{4}$ \( 1 + 81219577008 T + \)\(18\!\cdots\!82\)\( T^{2} + 81219577008 p^{13} T^{3} + p^{26} T^{4} \)
67$D_{4}$ \( 1 - 229155633102 T + \)\(10\!\cdots\!31\)\( T^{2} - 229155633102 p^{13} T^{3} + p^{26} T^{4} \)
71$D_{4}$ \( 1 + 1161878914578 T + \)\(15\!\cdots\!67\)\( T^{2} + 1161878914578 p^{13} T^{3} + p^{26} T^{4} \)
73$D_{4}$ \( 1 - 456037317380 T - \)\(20\!\cdots\!50\)\( T^{2} - 456037317380 p^{13} T^{3} + p^{26} T^{4} \)
79$D_{4}$ \( 1 - 63811251988 p T + \)\(15\!\cdots\!98\)\( T^{2} - 63811251988 p^{14} T^{3} + p^{26} T^{4} \)
83$D_{4}$ \( 1 - 5610906244940 T + \)\(22\!\cdots\!30\)\( T^{2} - 5610906244940 p^{13} T^{3} + p^{26} T^{4} \)
89$D_{4}$ \( 1 + 3239330626042 T + \)\(31\!\cdots\!23\)\( T^{2} + 3239330626042 p^{13} T^{3} + p^{26} T^{4} \)
97$D_{4}$ \( 1 + 20541366120174 T + \)\(23\!\cdots\!23\)\( T^{2} + 20541366120174 p^{13} T^{3} + p^{26} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.95140131802103981201867073410, −14.87474104117674250529636473941, −13.99997988268235395646032530887, −13.76683523600346025627041116867, −12.25823527277493036412370418465, −11.91873527139603806770244825387, −10.85705119327016153745744460672, −10.66542434971115677112512000047, −9.531570423869047221233490692268, −9.013362933866213392521717675652, −8.217711651106328436347817370147, −8.185503035538439615815396270225, −7.27939813540220854995296396707, −6.39464833743860959886997294105, −5.11282740790511666289200509508, −4.21473712046343081174677530835, −2.75707556591662842592296823267, −2.40424951975807663527369403340, −1.24729011454642587542203679716, −0.988929169996361016477895706346, 0.988929169996361016477895706346, 1.24729011454642587542203679716, 2.40424951975807663527369403340, 2.75707556591662842592296823267, 4.21473712046343081174677530835, 5.11282740790511666289200509508, 6.39464833743860959886997294105, 7.27939813540220854995296396707, 8.185503035538439615815396270225, 8.217711651106328436347817370147, 9.013362933866213392521717675652, 9.531570423869047221233490692268, 10.66542434971115677112512000047, 10.85705119327016153745744460672, 11.91873527139603806770244825387, 12.25823527277493036412370418465, 13.76683523600346025627041116867, 13.99997988268235395646032530887, 14.87474104117674250529636473941, 14.95140131802103981201867073410

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