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Properties

Label 4-676000-1.1-c1e2-0-19
Degree $4$
Conductor $676000$
Sign $1$
Analytic cond. $43.1023$
Root an. cond. $2.56227$
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-s + 4-s + 5-s − 8-s + 2·9-s − 10-s + 2·13-s + 16-s + 12·17-s − 2·18-s + 20-s + 25-s − 2·26-s − 32-s − 12·34-s + 2·36-s + 20·37-s − 40-s + 2·45-s + 2·49-s − 50-s + 2·52-s + 8·61-s + 64-s + 2·65-s + 12·68-s − 2·72-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s + 2/3·9-s − 0.316·10-s + 0.554·13-s + 1/4·16-s + 2.91·17-s − 0.471·18-s + 0.223·20-s + 1/5·25-s − 0.392·26-s − 0.176·32-s − 2.05·34-s + 1/3·36-s + 3.28·37-s − 0.158·40-s + 0.298·45-s + 2/7·49-s − 0.141·50-s + 0.277·52-s + 1.02·61-s + 1/8·64-s + 0.248·65-s + 1.45·68-s − 0.235·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(676000\)    =    \(2^{5} \cdot 5^{3} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(43.1023\)
Root analytic conductor: \(2.56227\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 676000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.152882762\)
\(L(\frac12)\) \(\approx\) \(2.152882762\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5$C_1$ \( 1 - T \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.19.a_o
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.37.au_gs
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.43.a_abi
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.59.a_ck
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.ai_dy
67$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.67.a_w
71$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.71.a_ba
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.83.a_aby
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.ae_cc
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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

−8.188213835484741838379019034374, −7.906651190544632703319963507741, −7.63173525560074970155899983312, −7.07689529153056017366769268531, −6.62775562107752188584675775872, −5.98708778541359078123163804732, −5.72815273951368910547380197593, −5.35553332391136858705833592166, −4.54261471258447342957739948742, −4.05885592794284975938981119028, −3.40056417671944035392004617652, −2.90723599786168770057771565760, −2.24769647416774890006733188100, −1.23256315609033343339490834105, −1.03402461410424602848405662774, 1.03402461410424602848405662774, 1.23256315609033343339490834105, 2.24769647416774890006733188100, 2.90723599786168770057771565760, 3.40056417671944035392004617652, 4.05885592794284975938981119028, 4.54261471258447342957739948742, 5.35553332391136858705833592166, 5.72815273951368910547380197593, 5.98708778541359078123163804732, 6.62775562107752188584675775872, 7.07689529153056017366769268531, 7.63173525560074970155899983312, 7.906651190544632703319963507741, 8.188213835484741838379019034374

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