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Properties

Label 4-387200-1.1-c1e2-0-6
Degree $4$
Conductor $387200$
Sign $-1$
Analytic cond. $24.6882$
Root an. cond. $2.22906$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 8·7-s + 5·9-s − 4·25-s + 8·35-s + 16·43-s − 5·45-s + 34·49-s − 40·63-s + 16·81-s + 2·89-s − 8·107-s − 11·121-s + 9·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 32·175-s + 179-s + ⋯
L(s)  = 1  − 0.447·5-s − 3.02·7-s + 5/3·9-s − 4/5·25-s + 1.35·35-s + 2.43·43-s − 0.745·45-s + 34/7·49-s − 5.03·63-s + 16/9·81-s + 0.211·89-s − 0.773·107-s − 121-s + 0.804·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + 2.41·175-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(387200\)    =    \(2^{7} \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(24.6882\)
Root analytic conductor: \(2.22906\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 387200,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_2$ \( 1 + T + p T^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.3.a_af
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.23.a_abl
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.31.a_at
37$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.37.a_bx
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.41.a_ck
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.47.a_ada
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
59$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.59.a_cr
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.67.a_an
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.71.a_cj
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.a_fa
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.89.ac_gx
97$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.97.a_hl
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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

−8.510945746938236952063656017949, −7.76072531931273225323728176794, −7.43283588566094356954993982923, −7.01116515034650669628952043875, −6.63892584751836306565562763174, −6.14694828463787054199667199738, −5.88520310320994654143191550490, −5.07816524322177327984650421499, −4.26829128202728402629612811224, −3.91210863693790243054578893983, −3.56509892243236173094785516534, −2.88721059272235607002443522146, −2.29327824632832445827948559491, −1.06869591694087430841968556241, 0, 1.06869591694087430841968556241, 2.29327824632832445827948559491, 2.88721059272235607002443522146, 3.56509892243236173094785516534, 3.91210863693790243054578893983, 4.26829128202728402629612811224, 5.07816524322177327984650421499, 5.88520310320994654143191550490, 6.14694828463787054199667199738, 6.63892584751836306565562763174, 7.01116515034650669628952043875, 7.43283588566094356954993982923, 7.76072531931273225323728176794, 8.510945746938236952063656017949

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