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Properties

Label 4-837e2-1.1-c1e2-0-8
Degree $4$
Conductor $700569$
Sign $1$
Analytic cond. $44.6688$
Root an. cond. $2.58524$
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  + 2·4-s + 7-s + 7·19-s + 5·25-s + 2·28-s − 4·31-s + 7·49-s − 8·64-s + 10·67-s + 14·76-s + 19·97-s + 10·100-s + 13·103-s − 2·109-s − 22·121-s − 8·124-s + 127-s + 131-s + 7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯
L(s)  = 1  + 4-s + 0.377·7-s + 1.60·19-s + 25-s + 0.377·28-s − 0.718·31-s + 49-s − 64-s + 1.22·67-s + 1.60·76-s + 1.92·97-s + 100-s + 1.28·103-s − 0.191·109-s − 2·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.606·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.0769·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(700569\)    =    \(3^{6} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(44.6688\)
Root analytic conductor: \(2.58524\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 700569,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.912705256\)
\(L(\frac12)\) \(\approx\) \(2.912705256\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
31$C_2$ \( 1 + 4 T + p T^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.2.a_ac
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.ab_ag
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.a_b
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ah_be
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.a_abv
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.a_er
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.67.ak_gd
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.71.a_act
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.a_dt
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.79.a_afb
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.97.at_ke
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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

−8.253855500104420877186555870182, −7.67037139196828632860862122573, −7.48170101090988193337775076716, −6.97412284185805124905273757444, −6.66356519840459564016076631466, −6.07200079967964012472909585825, −5.62058372531395570960582579568, −5.09057675998369784477455638026, −4.75871322890928829528567186112, −3.97309423489568419329525083015, −3.43892948080391911601485533551, −2.87338977216832978632899858078, −2.32902075859861711955554627012, −1.65838169163521806742589097644, −0.874146279885474329311284138395, 0.874146279885474329311284138395, 1.65838169163521806742589097644, 2.32902075859861711955554627012, 2.87338977216832978632899858078, 3.43892948080391911601485533551, 3.97309423489568419329525083015, 4.75871322890928829528567186112, 5.09057675998369784477455638026, 5.62058372531395570960582579568, 6.07200079967964012472909585825, 6.66356519840459564016076631466, 6.97412284185805124905273757444, 7.48170101090988193337775076716, 7.67037139196828632860862122573, 8.253855500104420877186555870182

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