↑

Properties

Label 4-431244-1.1-c1e2-0-2
Degree $4$
Conductor $431244$
Sign $1$
Analytic cond. $27.4964$
Root an. cond. $2.28991$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 11-s + 16-s − 10·25-s − 8·31-s + 4·37-s − 44-s + 24·47-s − 10·49-s − 24·53-s + 24·59-s + 64-s + 16·67-s + 24·71-s + 4·97-s − 10·100-s + 16·103-s + 24·113-s + 121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 4·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.301·11-s + 1/4·16-s − 2·25-s − 1.43·31-s + 0.657·37-s − 0.150·44-s + 3.50·47-s − 1.42·49-s − 3.29·53-s + 3.12·59-s + 1/8·64-s + 1.95·67-s + 2.84·71-s + 0.406·97-s − 100-s + 1.57·103-s + 2.25·113-s + 1/11·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.328·148-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431244 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(431244\)    =    \(2^{2} \cdot 3^{4} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(27.4964\)
Root analytic conductor: \(2.28991\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 431244,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.836860576\)
\(L(\frac12)\) \(\approx\) \(1.836860576\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
11$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.47.ay_je
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.53.y_jq
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.a_aby
79$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.79.a_fy
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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

−8.329295392613665337579164470522, −8.262955010043880759821963787460, −7.61495866909191069709821054220, −7.29731250016140642032626805500, −6.85457054556699367962615309050, −6.04977512526916981733349434592, −6.03837711275371548392107302554, −5.28004968616744444473082066349, −4.94712570992833096255379440889, −4.01426444342608389151630027274, −3.78216905129380833725816829242, −3.11110186957338669233243757307, −2.12387935125240219535618583660, −2.06078284868063780415053686864, −0.72040081483209956050869656181, 0.72040081483209956050869656181, 2.06078284868063780415053686864, 2.12387935125240219535618583660, 3.11110186957338669233243757307, 3.78216905129380833725816829242, 4.01426444342608389151630027274, 4.94712570992833096255379440889, 5.28004968616744444473082066349, 6.03837711275371548392107302554, 6.04977512526916981733349434592, 6.85457054556699367962615309050, 7.29731250016140642032626805500, 7.61495866909191069709821054220, 8.262955010043880759821963787460, 8.329295392613665337579164470522

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