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Properties

Label 4-311904-1.1-c1e2-0-12
Degree $4$
Conductor $311904$
Sign $1$
Analytic cond. $19.8872$
Root an. cond. $2.11175$
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-s + 3-s − 4-s + 6-s + 4·7-s − 3·8-s + 9-s − 12-s + 4·14-s − 16-s + 18-s + 4·19-s + 4·21-s − 3·24-s + 10·25-s + 27-s − 4·28-s − 4·29-s + 5·32-s − 36-s + 4·38-s − 8·41-s + 4·42-s + 8·43-s − 48-s + 2·49-s + 10·50-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.288·12-s + 1.06·14-s − 1/4·16-s + 0.235·18-s + 0.917·19-s + 0.872·21-s − 0.612·24-s + 2·25-s + 0.192·27-s − 0.755·28-s − 0.742·29-s + 0.883·32-s − 1/6·36-s + 0.648·38-s − 1.24·41-s + 0.617·42-s + 1.21·43-s − 0.144·48-s + 2/7·49-s + 1.41·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(311904\)    =    \(2^{5} \cdot 3^{3} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(19.8872\)
Root analytic conductor: \(2.11175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 311904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.301589017\)
\(L(\frac12)\) \(\approx\) \(3.301589017\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
3$C_1$ \( 1 - T \)
19$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.7.ae_o
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.e_bu
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.31.a_aby
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.i_ck
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.43.ai_di
47$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.47.a_ac
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.ae_dq
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.am_dq
67$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.67.a_abq
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.m_eo
79$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.79.a_ac
83$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.83.a_as
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ai_dq
97$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.97.a_ade
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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

−8.698939056102647300872667163861, −8.441340065360205641188252518872, −7.978503853500202863289244441111, −7.32870315235511902958617857536, −7.13592049138550436112422074744, −6.36274198426465621472588450557, −5.75050995403316761441828087498, −5.20359902733101892504815780523, −4.92750562920968109525010541081, −4.42854748229958997474130460290, −3.89170606720577417272709640939, −3.21092788071506759254105051118, −2.73994196620849392373383277329, −1.83115779446539369074597003093, −1.00697571581039880902660971444, 1.00697571581039880902660971444, 1.83115779446539369074597003093, 2.73994196620849392373383277329, 3.21092788071506759254105051118, 3.89170606720577417272709640939, 4.42854748229958997474130460290, 4.92750562920968109525010541081, 5.20359902733101892504815780523, 5.75050995403316761441828087498, 6.36274198426465621472588450557, 7.13592049138550436112422074744, 7.32870315235511902958617857536, 7.978503853500202863289244441111, 8.441340065360205641188252518872, 8.698939056102647300872667163861

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