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Properties

Label 4-266240-1.1-c1e2-0-19
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·9-s − 13-s − 2·17-s − 4·25-s + 4·29-s − 8·37-s − 2·41-s − 2·45-s − 12·49-s + 8·53-s − 4·61-s − 65-s − 16·73-s − 5·81-s − 2·85-s + 8·89-s − 14·97-s + 2·109-s + 2·113-s + 2·117-s − 14·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.447·5-s − 2/3·9-s − 0.277·13-s − 0.485·17-s − 4/5·25-s + 0.742·29-s − 1.31·37-s − 0.312·41-s − 0.298·45-s − 1.71·49-s + 1.09·53-s − 0.512·61-s − 0.124·65-s − 1.87·73-s − 5/9·81-s − 0.216·85-s + 0.847·89-s − 1.42·97-s + 0.191·109-s + 0.188·113-s + 0.184·117-s − 1.27·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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$ \( ( 1 - T )( 1 + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.7.a_m
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.c_k
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.ae_bu
31$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.31.a_ag
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.i_di
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.c_de
43$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.43.a_bu
47$C_2^2$ \( 1 + 84 T^{2} + p^{2} T^{4} \) 2.47.a_dg
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.53.ai_eo
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.59.a_ady
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.e_eg
67$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \) 2.67.a_ado
71$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.71.a_aeg
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.q_gs
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.79.a_be
83$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \) 2.83.a_ace
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.89.ai_hi
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.o_ik
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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

−8.716132948436351731207458177640, −8.284794677196306257121639499050, −7.77772094883558885874540585711, −7.23157919662931160066439247666, −6.71742685433549115450815519607, −6.30252919297195706938972691353, −5.73274582051526831821288420541, −5.33486506239524666979553343161, −4.74339430379872145198254856333, −4.20681787675333551882844522092, −3.45520348996839070906463815446, −2.89282382742866434830024397922, −2.21096894291985121633293072059, −1.47214477447302869375249291572, 0, 1.47214477447302869375249291572, 2.21096894291985121633293072059, 2.89282382742866434830024397922, 3.45520348996839070906463815446, 4.20681787675333551882844522092, 4.74339430379872145198254856333, 5.33486506239524666979553343161, 5.73274582051526831821288420541, 6.30252919297195706938972691353, 6.71742685433549115450815519607, 7.23157919662931160066439247666, 7.77772094883558885874540585711, 8.284794677196306257121639499050, 8.716132948436351731207458177640

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