Properties

Label 4-156800-1.1-c1e2-0-36
Degree $4$
Conductor $156800$
Sign $-1$
Analytic cond. $9.99770$
Root an. cond. $1.77817$
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  − 2-s + 4-s + 2·7-s − 8-s − 2·9-s − 2·14-s + 16-s − 6·17-s + 2·18-s + 25-s + 2·28-s − 8·31-s − 32-s + 6·34-s − 2·36-s − 6·41-s − 6·47-s + 3·49-s − 50-s − 2·56-s + 8·62-s − 4·63-s + 64-s − 6·68-s + 2·72-s + 4·73-s − 20·79-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.471·18-s + 1/5·25-s + 0.377·28-s − 1.43·31-s − 0.176·32-s + 1.02·34-s − 1/3·36-s − 0.937·41-s − 0.875·47-s + 3/7·49-s − 0.141·50-s − 0.267·56-s + 1.01·62-s − 0.503·63-s + 1/8·64-s − 0.727·68-s + 0.235·72-s + 0.468·73-s − 2.25·79-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: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 156800,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2$C_1$ \( 1 + T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
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$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.900967404214814360155728720045, −8.595391550004089428257614214162, −8.175114563020558380296865729629, −7.69476575482621482508469057464, −6.94526069192045777044790762582, −6.86171931732588786239234570025, −6.03311756257113652450391513724, −5.57607001516633241815407220328, −4.97470123691061072321941481938, −4.40871030244508749874393512611, −3.68296580336228982924419474549, −2.91521048978382309590202053643, −2.18735775250471208006222515460, −1.50991460483536013897516168917, 0, 1.50991460483536013897516168917, 2.18735775250471208006222515460, 2.91521048978382309590202053643, 3.68296580336228982924419474549, 4.40871030244508749874393512611, 4.97470123691061072321941481938, 5.57607001516633241815407220328, 6.03311756257113652450391513724, 6.86171931732588786239234570025, 6.94526069192045777044790762582, 7.69476575482621482508469057464, 8.175114563020558380296865729629, 8.595391550004089428257614214162, 8.900967404214814360155728720045

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