Properties

Label 4-3920e2-1.1-c1e2-0-7
Degree $4$
Conductor $15366400$
Sign $1$
Analytic cond. $979.774$
Root an. cond. $5.59476$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s − 9-s − 11-s − 5·13-s + 2·15-s + 5·17-s − 6·19-s + 2·23-s + 3·25-s + 29-s + 33-s + 12·37-s + 5·39-s − 2·41-s − 10·43-s + 2·45-s − 5·47-s − 5·51-s − 2·53-s + 2·55-s + 6·57-s − 8·59-s − 6·61-s + 10·65-s − 4·67-s − 2·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 1/3·9-s − 0.301·11-s − 1.38·13-s + 0.516·15-s + 1.21·17-s − 1.37·19-s + 0.417·23-s + 3/5·25-s + 0.185·29-s + 0.174·33-s + 1.97·37-s + 0.800·39-s − 0.312·41-s − 1.52·43-s + 0.298·45-s − 0.729·47-s − 0.700·51-s − 0.274·53-s + 0.269·55-s + 0.794·57-s − 1.04·59-s − 0.768·61-s + 1.24·65-s − 0.488·67-s − 0.240·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15366400\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(979.774\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 15366400,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 6 T - 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 9 T + 174 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 6 T + 170 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 9 T + 108 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
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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

−8.122152840363497163874171911188, −7.74004540107140700552565840350, −7.66961860114679651126489267335, −7.41470083505652831045822138805, −6.60300811048339436766613681731, −6.54328572484194140102944955130, −6.13108024965993165777572245944, −5.67990010127077367454731080622, −5.17958190868467841924170862926, −4.85598705645500092368332686501, −4.62757011875040553518821123890, −4.19111071326445571508763274720, −3.49699879177161230680916513578, −3.35545520121437657410584336392, −2.59403383504048912179386416757, −2.48399588045007103902069471250, −1.62049698385777854730253003062, −1.02973534269655970867892478433, 0, 0, 1.02973534269655970867892478433, 1.62049698385777854730253003062, 2.48399588045007103902069471250, 2.59403383504048912179386416757, 3.35545520121437657410584336392, 3.49699879177161230680916513578, 4.19111071326445571508763274720, 4.62757011875040553518821123890, 4.85598705645500092368332686501, 5.17958190868467841924170862926, 5.67990010127077367454731080622, 6.13108024965993165777572245944, 6.54328572484194140102944955130, 6.60300811048339436766613681731, 7.41470083505652831045822138805, 7.66961860114679651126489267335, 7.74004540107140700552565840350, 8.122152840363497163874171911188

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