Properties

Label 4-578000-1.1-c1e2-0-0
Degree $4$
Conductor $578000$
Sign $1$
Analytic cond. $36.8537$
Root an. cond. $2.46388$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 5-s + 4·7-s + 6·9-s + 4·15-s − 6·17-s − 8·19-s − 16·21-s + 12·23-s + 25-s + 4·27-s − 4·35-s + 4·37-s − 6·45-s − 2·49-s + 24·51-s + 32·57-s + 24·59-s + 24·63-s − 48·69-s + 4·73-s − 4·75-s − 37·81-s + 6·85-s − 12·89-s + 8·95-s + 4·97-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.447·5-s + 1.51·7-s + 2·9-s + 1.03·15-s − 1.45·17-s − 1.83·19-s − 3.49·21-s + 2.50·23-s + 1/5·25-s + 0.769·27-s − 0.676·35-s + 0.657·37-s − 0.894·45-s − 2/7·49-s + 3.36·51-s + 4.23·57-s + 3.12·59-s + 3.02·63-s − 5.77·69-s + 0.468·73-s − 0.461·75-s − 4.11·81-s + 0.650·85-s − 1.27·89-s + 0.820·95-s + 0.406·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 578000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(578000\)    =    \(2^{4} \cdot 5^{3} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(36.8537\)
Root analytic conductor: \(2.46388\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 578000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5768249526\)
\(L(\frac12)\) \(\approx\) \(0.5768249526\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5$C_1$ \( 1 + T \)
17$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.552217550781204179646223792184, −8.058415744982227744304745377698, −7.31794529895409889634812950719, −6.82713425889957194654989882337, −6.57891116465648258947670054106, −6.24314171685151676770356485369, −5.39532675004449418324120019391, −5.29055530149571575322692101844, −4.78130792717525308450176413839, −4.45683675746786112731591005065, −3.98648408873839658165348001528, −2.87939248236646346937923899702, −2.23497397622765245157605400813, −1.28472904085539070234156118632, −0.49512374452997244826071619331, 0.49512374452997244826071619331, 1.28472904085539070234156118632, 2.23497397622765245157605400813, 2.87939248236646346937923899702, 3.98648408873839658165348001528, 4.45683675746786112731591005065, 4.78130792717525308450176413839, 5.29055530149571575322692101844, 5.39532675004449418324120019391, 6.24314171685151676770356485369, 6.57891116465648258947670054106, 6.82713425889957194654989882337, 7.31794529895409889634812950719, 8.058415744982227744304745377698, 8.552217550781204179646223792184

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