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Properties

Label 4-555579-1.1-c1e2-0-6
Degree $4$
Conductor $555579$
Sign $-1$
Analytic cond. $35.4241$
Root an. cond. $2.43963$
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  − 4·4-s + 2·7-s + 12·16-s − 19-s + 25-s − 8·28-s − 2·43-s − 11·49-s + 2·61-s − 32·64-s − 14·73-s + 4·76-s − 4·100-s + 24·112-s + 13·121-s + 127-s + 131-s − 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 8·172-s + ⋯
L(s)  = 1  − 2·4-s + 0.755·7-s + 3·16-s − 0.229·19-s + 1/5·25-s − 1.51·28-s − 0.304·43-s − 1.57·49-s + 0.256·61-s − 4·64-s − 1.63·73-s + 0.458·76-s − 2/5·100-s + 2.26·112-s + 1.18·121-s + 0.0887·127-s + 0.0873·131-s − 0.173·133-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 + 0.609·172-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(555579\)    =    \(3^{4} \cdot 19^{3}\)
Sign: $-1$
Analytic conductor: \(35.4241\)
Root analytic conductor: \(2.43963\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 555579,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad3 \( 1 \)
19$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.2.a_e
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.7.ac_p
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.11.a_an
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.17.a_az
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.47.a_adh
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.a_abm
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.61.ac_et
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.73.o_hn
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.a_bi
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.a_fa
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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

−8.277559310587674810318806928876, −8.006016183793790414655346871072, −7.47825363876561247016541483864, −6.99741904848666774601281231664, −6.17133498278874380962465440599, −5.88428262888180354888228435076, −5.18706082399202763596939178143, −4.91152259450916259989852452845, −4.51412293203567888487455025354, −4.00777582092682802237463462933, −3.47293456998714688357220616241, −2.88232421944290003280516356221, −1.80630419511795802992573497943, −1.07714880552781060574578433716, 0, 1.07714880552781060574578433716, 1.80630419511795802992573497943, 2.88232421944290003280516356221, 3.47293456998714688357220616241, 4.00777582092682802237463462933, 4.51412293203567888487455025354, 4.91152259450916259989852452845, 5.18706082399202763596939178143, 5.88428262888180354888228435076, 6.17133498278874380962465440599, 6.99741904848666774601281231664, 7.47825363876561247016541483864, 8.006016183793790414655346871072, 8.277559310587674810318806928876

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