↑

Properties

Label 4-220e2-1.1-c1e2-0-7
Degree $4$
Conductor $48400$
Sign $-1$
Analytic cond. $3.08602$
Root an. cond. $1.32540$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 2·5-s + 4·16-s + 4·20-s − 25-s − 10·37-s − 14·49-s − 10·53-s − 8·64-s − 8·80-s − 9·81-s − 32·89-s + 10·97-s + 2·100-s + 30·113-s − 11·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 20·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 4-s − 0.894·5-s + 16-s + 0.894·20-s − 1/5·25-s − 1.64·37-s − 2·49-s − 1.37·53-s − 64-s − 0.894·80-s − 81-s − 3.39·89-s + 1.01·97-s + 1/5·100-s + 2.82·113-s − 121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.64·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-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: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 48400,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
13$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \) 2.13.a_ay
17$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.17.a_aq
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 + p^{2} T^{4} \) 2.23.a_a
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.k_by
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2^2$ \( 1 + p^{2} T^{4} \) 2.47.a_a
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.k_by
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.a_aw
67$C_2^2$ \( 1 + p^{2} T^{4} \) 2.67.a_a
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2^2$ \( 1 + 96 T^{2} + p^{2} T^{4} \) 2.73.a_ds
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.89.bg_qs
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.ak_by
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

−9.852024340235122833741391518755, −9.386259163176997657635396323614, −8.674808381720954475854505554430, −8.442202918887683905010377489369, −7.84790509970991130198178576956, −7.41180165929233602198035370898, −6.73872744596787361913210667473, −6.06787051419613740496874700112, −5.36461841713189340194511092763, −4.80941409969147688358064144431, −4.24439961947683773012575777830, −3.59443937704488831719026983493, −3.03118334319369076677172811252, −1.60178350478021375259511122569, 0, 1.60178350478021375259511122569, 3.03118334319369076677172811252, 3.59443937704488831719026983493, 4.24439961947683773012575777830, 4.80941409969147688358064144431, 5.36461841713189340194511092763, 6.06787051419613740496874700112, 6.73872744596787361913210667473, 7.41180165929233602198035370898, 7.84790509970991130198178576956, 8.442202918887683905010377489369, 8.674808381720954475854505554430, 9.386259163176997657635396323614, 9.852024340235122833741391518755

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