Properties

Label 4-1520e2-1.1-c1e2-0-15
Degree $4$
Conductor $2310400$
Sign $1$
Analytic cond. $147.313$
Root an. cond. $3.48385$
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  − 2·5-s + 6·9-s + 8·11-s + 2·19-s − 25-s + 12·29-s + 8·31-s − 20·41-s − 12·45-s + 10·49-s − 16·55-s + 4·61-s − 8·71-s + 8·79-s + 27·81-s + 4·89-s − 4·95-s + 48·99-s − 12·101-s − 12·109-s + 26·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + ⋯
L(s)  = 1  − 0.894·5-s + 2·9-s + 2.41·11-s + 0.458·19-s − 1/5·25-s + 2.22·29-s + 1.43·31-s − 3.12·41-s − 1.78·45-s + 10/7·49-s − 2.15·55-s + 0.512·61-s − 0.949·71-s + 0.900·79-s + 3·81-s + 0.423·89-s − 0.410·95-s + 4.82·99-s − 1.19·101-s − 1.14·109-s + 2.36·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.99·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2310400\)    =    \(2^{8} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(147.313\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1520} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2310400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.210145844\)
\(L(\frac12)\) \(\approx\) \(3.210145844\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \)
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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

−9.790974460603350459577491348606, −9.252687208463408578968196274943, −8.806023744950521977225630123010, −8.570691793726506884393445217455, −7.967209249590351443092100967014, −7.76788643969557361212284973950, −7.01972002348104482095709611608, −6.85927486385040849304503143387, −6.62479186634131935246910711123, −6.28470601196291565371436022421, −5.52323719735619434916334743381, −4.84547176461084993520862042114, −4.50484151717264425960352891776, −4.24941557729357117797711864981, −3.64562891955454834156087366130, −3.53009422507654082277128363168, −2.69093856588231215258699587545, −1.79689909352392986389703549877, −1.28701161864211654620757389450, −0.853342878682414649281281933757, 0.853342878682414649281281933757, 1.28701161864211654620757389450, 1.79689909352392986389703549877, 2.69093856588231215258699587545, 3.53009422507654082277128363168, 3.64562891955454834156087366130, 4.24941557729357117797711864981, 4.50484151717264425960352891776, 4.84547176461084993520862042114, 5.52323719735619434916334743381, 6.28470601196291565371436022421, 6.62479186634131935246910711123, 6.85927486385040849304503143387, 7.01972002348104482095709611608, 7.76788643969557361212284973950, 7.967209249590351443092100967014, 8.570691793726506884393445217455, 8.806023744950521977225630123010, 9.252687208463408578968196274943, 9.790974460603350459577491348606

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