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Properties

Label 4-220e2-1.1-c1e2-0-3
Degree $4$
Conductor $48400$
Sign $1$
Analytic cond. $3.08602$
Root an. cond. $1.32540$
Motivic weight $1$
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-s − 4-s + 2·5-s − 3·8-s − 6·9-s + 2·10-s + 4·13-s − 16-s + 12·17-s − 6·18-s − 2·20-s + 3·25-s + 4·26-s + 12·29-s + 5·32-s + 12·34-s + 6·36-s − 4·37-s − 6·40-s + 4·41-s − 12·45-s − 14·49-s + 3·50-s − 4·52-s − 4·53-s + 12·58-s − 20·61-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.06·8-s − 2·9-s + 0.632·10-s + 1.10·13-s − 1/4·16-s + 2.91·17-s − 1.41·18-s − 0.447·20-s + 3/5·25-s + 0.784·26-s + 2.22·29-s + 0.883·32-s + 2.05·34-s + 36-s − 0.657·37-s − 0.948·40-s + 0.624·41-s − 1.78·45-s − 2·49-s + 0.424·50-s − 0.554·52-s − 0.549·53-s + 1.57·58-s − 2.56·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(48400\)    =    \(2^{4} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(3.08602\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 48400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.770622110\)
\(L(\frac12)\) \(\approx\) \(1.770622110\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 - T + p T^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.a_aes
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.73.abc_ne
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−10.07588915028781859505118646541, −9.376036497146997809466572379928, −9.365267708430525132762162914792, −8.402071812243779349662792300606, −8.248642768475844318802620430882, −7.78041223670494609407513138154, −6.49908708730306794178412805091, −6.16821463973303244938343944630, −5.83555924118533390298503875284, −5.09766059609940450216711100862, −4.97617253141282559040022801022, −3.60799086718076388676488423337, −3.25423179226253980082861279896, −2.69303075235727048947048138558, −1.13555459814448309624758433433, 1.13555459814448309624758433433, 2.69303075235727048947048138558, 3.25423179226253980082861279896, 3.60799086718076388676488423337, 4.97617253141282559040022801022, 5.09766059609940450216711100862, 5.83555924118533390298503875284, 6.16821463973303244938343944630, 6.49908708730306794178412805091, 7.78041223670494609407513138154, 8.248642768475844318802620430882, 8.402071812243779349662792300606, 9.365267708430525132762162914792, 9.376036497146997809466572379928, 10.07588915028781859505118646541

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