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Properties

Label 4-220e2-1.1-c1e2-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 3·5-s + 5·9-s + 4·16-s − 6·20-s + 4·25-s − 10·36-s + 15·45-s − 14·49-s − 8·64-s + 12·80-s + 16·81-s + 18·89-s − 8·100-s − 11·121-s − 3·125-s + 127-s + 131-s + 137-s + 139-s + 20·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + ⋯
L(s)  = 1  − 4-s + 1.34·5-s + 5/3·9-s + 16-s − 1.34·20-s + 4/5·25-s − 5/3·36-s + 2.23·45-s − 2·49-s − 64-s + 1.34·80-s + 16/9·81-s + 1.90·89-s − 4/5·100-s − 121-s − 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-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: \(0\)
Selberg data: \((4,\ 48400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.554924035\)
\(L(\frac12)\) \(\approx\) \(1.554924035\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 3 T + p T^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.3.a_af
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.23.a_bj
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.a_bl
37$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.a_z
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.59.a_aed
61$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.61.a_aes
67$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.67.a_bj
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.71.a_fd
73$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.73.a_fq
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$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.89.as_jz
97$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.97.a_adr
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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

−9.994887975572103161857584022863, −9.704139309985122251087028314551, −9.206399605952974334275950360362, −8.826540164106492928979530754737, −7.980410044851693878181118821935, −7.66060479799340726988522517497, −6.82806327172260454160823115639, −6.42132685219677291431212929721, −5.76409605497127093972946940231, −5.09801796917653691051813939785, −4.67008852243717285152766977044, −4.00700244518755889615568073003, −3.23704798912274318369430972836, −2.06260117355070567414944378858, −1.27542971164956686515729630257, 1.27542971164956686515729630257, 2.06260117355070567414944378858, 3.23704798912274318369430972836, 4.00700244518755889615568073003, 4.67008852243717285152766977044, 5.09801796917653691051813939785, 5.76409605497127093972946940231, 6.42132685219677291431212929721, 6.82806327172260454160823115639, 7.66060479799340726988522517497, 7.980410044851693878181118821935, 8.826540164106492928979530754737, 9.206399605952974334275950360362, 9.704139309985122251087028314551, 9.994887975572103161857584022863

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