↑

Properties

Label 4-266240-1.1-c1e2-0-13
Degree $4$
Conductor $266240$
Sign $-1$
Analytic cond. $16.9756$
Root an. cond. $2.02981$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 2·9-s + 3·13-s + 2·17-s + 8·25-s − 20·29-s + 8·37-s + 2·41-s + 6·45-s + 8·49-s + 4·61-s − 9·65-s − 8·73-s − 5·81-s − 6·85-s − 2·97-s − 10·109-s − 18·113-s − 6·117-s + 10·121-s − 16·125-s + 127-s + 131-s + 137-s + 139-s + 60·145-s + 149-s + ⋯
L(s)  = 1  − 1.34·5-s − 2/3·9-s + 0.832·13-s + 0.485·17-s + 8/5·25-s − 3.71·29-s + 1.31·37-s + 0.312·41-s + 0.894·45-s + 8/7·49-s + 0.512·61-s − 1.11·65-s − 0.936·73-s − 5/9·81-s − 0.650·85-s − 0.203·97-s − 0.957·109-s − 1.69·113-s − 0.554·117-s + 0.909·121-s − 1.43·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.98·145-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 266240 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266240 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(266240\)    =    \(2^{12} \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(16.9756\)
Root analytic conductor: \(2.02981\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 266240,\ (\ :1/2, 1/2),\ -1)\)

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_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.7.a_ai
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ac_bi
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.19.a_ag
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.23.a_o
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.29.u_gc
31$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.31.a_by
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.ai_di
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.ac_abm
43$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.43.a_aba
47$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.47.a_aq
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
59$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.59.a_abm
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.67.a_ai
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.a_bq
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.i_gc
79$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.79.a_ck
83$C_2^2$ \( 1 + 148 T^{2} + p^{2} T^{4} \) 2.83.a_fs
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.c_hm
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

−8.597546963561134116897121385129, −8.210860680750116885551633292445, −7.65153076271034558381552312654, −7.43021441500417435699863960215, −6.95233788837970438987688029902, −6.19301562272761105771439384273, −5.69231112475036442582395165400, −5.41617102700533618632351567534, −4.58156222598047261147202852253, −3.92813147181408605164126820258, −3.71896851050457302397521622566, −3.06297671815207034614407867548, −2.28615710279814462741397780660, −1.20562388555508800722180889729, 0, 1.20562388555508800722180889729, 2.28615710279814462741397780660, 3.06297671815207034614407867548, 3.71896851050457302397521622566, 3.92813147181408605164126820258, 4.58156222598047261147202852253, 5.41617102700533618632351567534, 5.69231112475036442582395165400, 6.19301562272761105771439384273, 6.95233788837970438987688029902, 7.43021441500417435699863960215, 7.65153076271034558381552312654, 8.210860680750116885551633292445, 8.597546963561134116897121385129

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