Properties

Label 4-480e2-1.1-c1e2-0-35
Degree $4$
Conductor $230400$
Sign $1$
Analytic cond. $14.6905$
Root an. cond. $1.95775$
Motivic weight $1$
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  + 2·3-s + 2·5-s + 9-s + 4·15-s + 8·19-s + 12·23-s + 3·25-s − 4·27-s − 12·29-s + 20·43-s + 2·45-s − 12·47-s − 10·49-s + 12·53-s + 16·57-s − 4·67-s + 24·69-s − 24·71-s + 4·73-s + 6·75-s − 11·81-s − 24·87-s + 16·95-s + 4·97-s − 12·101-s + 24·115-s − 22·121-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 1/3·9-s + 1.03·15-s + 1.83·19-s + 2.50·23-s + 3/5·25-s − 0.769·27-s − 2.22·29-s + 3.04·43-s + 0.298·45-s − 1.75·47-s − 1.42·49-s + 1.64·53-s + 2.11·57-s − 0.488·67-s + 2.88·69-s − 2.84·71-s + 0.468·73-s + 0.692·75-s − 1.22·81-s − 2.57·87-s + 1.64·95-s + 0.406·97-s − 1.19·101-s + 2.23·115-s − 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(230400\)    =    \(2^{10} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(14.6905\)
Root analytic conductor: \(1.95775\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 230400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.256635373\)
\(L(\frac12)\) \(\approx\) \(3.256635373\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−9.039858624083400649706600435206, −8.847850276232423443211930888771, −7.87489459574230399033328236522, −7.74818737522785607031159368119, −7.09464940360169401329405651291, −6.88656092970954199609923322632, −5.90138678053477774939394585997, −5.59481538925271546647259599831, −5.17606419420266207753384183426, −4.45194901772205251160909130654, −3.70610300459195515417889608627, −3.01436611750632173441455711709, −2.84705552122166969163585152299, −1.91202694614300604029027352933, −1.16915866227454488376692012454, 1.16915866227454488376692012454, 1.91202694614300604029027352933, 2.84705552122166969163585152299, 3.01436611750632173441455711709, 3.70610300459195515417889608627, 4.45194901772205251160909130654, 5.17606419420266207753384183426, 5.59481538925271546647259599831, 5.90138678053477774939394585997, 6.88656092970954199609923322632, 7.09464940360169401329405651291, 7.74818737522785607031159368119, 7.87489459574230399033328236522, 8.847850276232423443211930888771, 9.039858624083400649706600435206

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