↑

Properties

Label 4-630e2-1.1-c1e2-0-30
Degree $4$
Conductor $396900$
Sign $1$
Analytic cond. $25.3066$
Root an. cond. $2.24289$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 2·5-s − 2·7-s + 16-s + 4·17-s + 2·20-s + 3·25-s − 2·28-s − 4·35-s + 8·37-s − 8·41-s + 16·47-s − 3·49-s − 12·59-s + 64-s − 8·67-s + 4·68-s + 8·79-s + 2·80-s + 8·83-s + 8·85-s + 3·100-s + 4·101-s + 20·109-s − 2·112-s − 8·119-s − 2·121-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.894·5-s − 0.755·7-s + 1/4·16-s + 0.970·17-s + 0.447·20-s + 3/5·25-s − 0.377·28-s − 0.676·35-s + 1.31·37-s − 1.24·41-s + 2.33·47-s − 3/7·49-s − 1.56·59-s + 1/8·64-s − 0.977·67-s + 0.485·68-s + 0.900·79-s + 0.223·80-s + 0.878·83-s + 0.867·85-s + 3/10·100-s + 0.398·101-s + 1.91·109-s − 0.188·112-s − 0.733·119-s − 0.181·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(396900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(25.3066\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 396900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.391056566\)
\(L(\frac12)\) \(\approx\) \(2.391056566\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_2$ \( 1 + 2 T + p T^{2} \)
good11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.11.a_c
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ae_w
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.31.a_be
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.i_de
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.53.a_acg
59$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.59.m_fi
61$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \) 2.61.a_dy
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.i_di
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.71.a_aby
73$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \) 2.73.a_dq
79$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.ai_eg
83$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.83.ai_gk
89$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.89.a_gg
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.97.a_aby
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

−8.752865539694197495282709374954, −8.115608151697774147296468633778, −7.73353283250992751127139585752, −7.17769186970973697348487450487, −6.82346223695145956817788989826, −6.20740924360800917964256882554, −5.88510876406748555510794878056, −5.59612208991143241807754404223, −4.81813924281743708921791085547, −4.36059359273349356283752908379, −3.42166966104912437443060611100, −3.17294654500634554697426824407, −2.42971757765403906248727725628, −1.80836103764466593632998458586, −0.876315552158498252007017333656, 0.876315552158498252007017333656, 1.80836103764466593632998458586, 2.42971757765403906248727725628, 3.17294654500634554697426824407, 3.42166966104912437443060611100, 4.36059359273349356283752908379, 4.81813924281743708921791085547, 5.59612208991143241807754404223, 5.88510876406748555510794878056, 6.20740924360800917964256882554, 6.82346223695145956817788989826, 7.17769186970973697348487450487, 7.73353283250992751127139585752, 8.115608151697774147296468633778, 8.752865539694197495282709374954

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