Properties

Label 4-3920e2-1.1-c0e2-0-5
Degree $4$
Conductor $15366400$
Sign $1$
Analytic cond. $3.82724$
Root an. cond. $1.39869$
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  + 3-s + 5-s + 9-s − 11-s + 2·13-s + 15-s − 17-s + 2·27-s − 2·29-s − 33-s + 2·39-s + 45-s + 47-s − 51-s − 55-s + 2·65-s − 4·71-s + 2·73-s − 79-s + 2·81-s + 4·83-s − 85-s − 2·87-s + 2·97-s − 99-s + 103-s + 109-s + ⋯
L(s)  = 1  + 3-s + 5-s + 9-s − 11-s + 2·13-s + 15-s − 17-s + 2·27-s − 2·29-s − 33-s + 2·39-s + 45-s + 47-s − 51-s − 55-s + 2·65-s − 4·71-s + 2·73-s − 79-s + 2·81-s + 4·83-s − 85-s − 2·87-s + 2·97-s − 99-s + 103-s + 109-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(15366400\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3.82724\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15366400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.828662815\)
\(L(\frac12)\) \(\approx\) \(2.828662815\)
\(L(1)\) 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 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good3$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
11$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
13$C_2$ \( ( 1 - T + T^{2} )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_2$ \( ( 1 + T + T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_1$ \( ( 1 + T )^{4} \)
73$C_2$ \( ( 1 - T + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
83$C_1$ \( ( 1 - T )^{4} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_2$ \( ( 1 - T + T^{2} )^{2} \)
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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

−8.733614783660548093633074642694, −8.652446336320477945843565088367, −8.212454549756463983408855911705, −7.65222500137946550544495172951, −7.46690627729885636446035196478, −7.17316743724677732856886375801, −6.46720430206765895279857520882, −6.18882148716012650976353155581, −6.07442689358017728000071771267, −5.52467628232886686018977425399, −5.00004235517666970918218422334, −4.78707212293264494304231723380, −3.97712806400171567523942674410, −3.97032340608221986595279972084, −3.27408788186626391917586658343, −3.04221483719978103522570925985, −2.24601110414887581424762709628, −2.13371016779172160361029019879, −1.59228024958307104524064297409, −0.927869125450152359380604797974, 0.927869125450152359380604797974, 1.59228024958307104524064297409, 2.13371016779172160361029019879, 2.24601110414887581424762709628, 3.04221483719978103522570925985, 3.27408788186626391917586658343, 3.97032340608221986595279972084, 3.97712806400171567523942674410, 4.78707212293264494304231723380, 5.00004235517666970918218422334, 5.52467628232886686018977425399, 6.07442689358017728000071771267, 6.18882148716012650976353155581, 6.46720430206765895279857520882, 7.17316743724677732856886375801, 7.46690627729885636446035196478, 7.65222500137946550544495172951, 8.212454549756463983408855911705, 8.652446336320477945843565088367, 8.733614783660548093633074642694

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