Properties

Label 4-156800-1.1-c1e2-0-5
Degree $4$
Conductor $156800$
Sign $1$
Analytic cond. $9.99770$
Root an. cond. $1.77817$
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  + 2-s + 4-s − 2·7-s + 8-s − 6·9-s − 2·14-s + 16-s + 4·17-s − 6·18-s + 25-s − 2·28-s + 16·31-s + 32-s + 4·34-s − 6·36-s + 4·41-s + 16·47-s + 3·49-s + 50-s − 2·56-s + 16·62-s + 12·63-s + 64-s + 4·68-s − 32·71-s − 6·72-s + 4·73-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 2·9-s − 0.534·14-s + 1/4·16-s + 0.970·17-s − 1.41·18-s + 1/5·25-s − 0.377·28-s + 2.87·31-s + 0.176·32-s + 0.685·34-s − 36-s + 0.624·41-s + 2.33·47-s + 3/7·49-s + 0.141·50-s − 0.267·56-s + 2.03·62-s + 1.51·63-s + 1/8·64-s + 0.485·68-s − 3.79·71-s − 0.707·72-s + 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(156800\)    =    \(2^{7} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(9.99770\)
Root analytic conductor: \(1.77817\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{156800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 156800,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.970166865\)
\(L(\frac12)\) \(\approx\) \(1.970166865\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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

−8.977646092372605556420891081568, −8.910824986105461683565850952843, −8.342925504546681427970604924895, −7.63810043151096799752774274091, −7.45420248729985818641828193491, −6.50661147582867110066747443111, −6.17616144092315208416665412995, −5.82344169953684002039972565231, −5.35297783463712406492518278139, −4.63923341949934829303628265619, −4.07505442753619333270812979660, −3.11732069850814337090071774581, −3.00381372509091040158022668795, −2.34423344984301921891298897662, −0.837608542604674414323310984226, 0.837608542604674414323310984226, 2.34423344984301921891298897662, 3.00381372509091040158022668795, 3.11732069850814337090071774581, 4.07505442753619333270812979660, 4.63923341949934829303628265619, 5.35297783463712406492518278139, 5.82344169953684002039972565231, 6.17616144092315208416665412995, 6.50661147582867110066747443111, 7.45420248729985818641828193491, 7.63810043151096799752774274091, 8.342925504546681427970604924895, 8.910824986105461683565850952843, 8.977646092372605556420891081568

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