Properties

Label 4-1960e2-1.1-c0e2-0-0
Degree $4$
Conductor $3841600$
Sign $1$
Analytic cond. $0.956811$
Root an. cond. $0.989023$
Motivic weight $0$
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 − 5-s − 8-s − 9-s − 10-s + 11-s − 2·13-s − 16-s − 18-s − 19-s + 22-s − 23-s − 2·26-s − 37-s − 38-s + 40-s + 2·41-s + 45-s − 46-s + 47-s − 53-s − 55-s + 2·59-s + 64-s + 2·65-s + 72-s − 74-s + ⋯
L(s)  = 1  + 2-s − 5-s − 8-s − 9-s − 10-s + 11-s − 2·13-s − 16-s − 18-s − 19-s + 22-s − 23-s − 2·26-s − 37-s − 38-s + 40-s + 2·41-s + 45-s − 46-s + 47-s − 53-s − 55-s + 2·59-s + 64-s + 2·65-s + 72-s − 74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3841600\)    =    \(2^{6} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.956811\)
Root analytic conductor: \(0.989023\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3841600,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7107458403\)
\(L(\frac12)\) \(\approx\) \(0.7107458403\)
\(L(1)\) 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$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good3$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 + T + T^{2} )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
41$C_2$ \( ( 1 - T + T^{2} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−9.903016701238506036072904180450, −9.040653047713604678570768425618, −8.825314246377564816052352640215, −8.447999485081896129564081944490, −7.87013434859997303068917552513, −7.70304075584915321926070795753, −7.14933828685915463819293228801, −6.79473770906631067789140861898, −6.19170417659035679578922542186, −6.07864227645811768284050192545, −5.29941768361267823946218681733, −5.26641287380979921991738051875, −4.41936262204184481569765927544, −4.40631370173048945845939277510, −3.72097064814655019970564345140, −3.65057539839967050742999293568, −2.64981295554159799151061999250, −2.60555079813967805201716209879, −1.87546853892448540342866567915, −0.49895080111073473926780434137, 0.49895080111073473926780434137, 1.87546853892448540342866567915, 2.60555079813967805201716209879, 2.64981295554159799151061999250, 3.65057539839967050742999293568, 3.72097064814655019970564345140, 4.40631370173048945845939277510, 4.41936262204184481569765927544, 5.26641287380979921991738051875, 5.29941768361267823946218681733, 6.07864227645811768284050192545, 6.19170417659035679578922542186, 6.79473770906631067789140861898, 7.14933828685915463819293228801, 7.70304075584915321926070795753, 7.87013434859997303068917552513, 8.447999485081896129564081944490, 8.825314246377564816052352640215, 9.040653047713604678570768425618, 9.903016701238506036072904180450

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