Properties

Label 4-800320-1.1-c1e2-0-0
Degree $4$
Conductor $800320$
Sign $1$
Analytic cond. $51.0290$
Root an. cond. $2.67272$
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  − 5-s + 9-s − 4·13-s + 17-s − 2·25-s − 8·29-s − 11·37-s − 41-s − 45-s + 10·49-s + 17·53-s + 6·61-s + 4·65-s − 8·73-s − 8·81-s − 85-s − 2·89-s + 15·97-s − 109-s − 5·113-s − 4·117-s − 3·121-s + 10·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.447·5-s + 1/3·9-s − 1.10·13-s + 0.242·17-s − 2/5·25-s − 1.48·29-s − 1.80·37-s − 0.156·41-s − 0.149·45-s + 10/7·49-s + 2.33·53-s + 0.768·61-s + 0.496·65-s − 0.936·73-s − 8/9·81-s − 0.108·85-s − 0.211·89-s + 1.52·97-s − 0.0957·109-s − 0.470·113-s − 0.369·117-s − 0.272·121-s + 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800320 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800320 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(800320\)    =    \(2^{6} \cdot 5 \cdot 41 \cdot 61\)
Sign: $1$
Analytic conductor: \(51.0290\)
Root analytic conductor: \(2.67272\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{800320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 800320,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.241743779\)
\(L(\frac12)\) \(\approx\) \(1.241743779\)
\(L(\frac{3}{2})\) 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$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 5 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 69 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 116 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−8.336537595081558363536543116360, −7.55968701280735602913297345958, −7.38088521355930822119241960331, −7.12112674986902485107562903479, −6.59774128279327782738959151801, −5.88113909531543415005029227612, −5.43354993935349498674674587861, −5.18917654756010778615059472905, −4.45123899867786347293233165561, −3.99053471079608384032698389401, −3.61321196000800542168560242805, −2.90464359196295271223677775255, −2.22942377040571515198940565646, −1.68138720270142987854904066464, −0.52818073907297871319586173929, 0.52818073907297871319586173929, 1.68138720270142987854904066464, 2.22942377040571515198940565646, 2.90464359196295271223677775255, 3.61321196000800542168560242805, 3.99053471079608384032698389401, 4.45123899867786347293233165561, 5.18917654756010778615059472905, 5.43354993935349498674674587861, 5.88113909531543415005029227612, 6.59774128279327782738959151801, 7.12112674986902485107562903479, 7.38088521355930822119241960331, 7.55968701280735602913297345958, 8.336537595081558363536543116360

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