Properties

Label 4-720e2-1.1-c1e2-0-31
Degree $4$
Conductor $518400$
Sign $1$
Analytic cond. $33.0536$
Root an. cond. $2.39775$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 12·13-s − 25-s − 12·37-s + 2·49-s + 12·61-s + 28·73-s − 4·97-s + 20·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 3.32·13-s − 1/5·25-s − 1.97·37-s + 2/7·49-s + 1.53·61-s + 3.27·73-s − 0.406·97-s + 1.91·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(518400\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(33.0536\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{518400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 518400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.321057923\)
\(L(\frac12)\) \(\approx\) \(2.321057923\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + T^{2} \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 174 T^{2} + p^{2} T^{4} \)
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

−8.580277831204113965788041851449, −8.188878433695120589679764983709, −7.71581291411964917884463964237, −6.99088542322115733938972134956, −6.57774176915863429748029454892, −6.31251491568464392148089374759, −5.65309758648045018186085625163, −5.42232139811458997298984942476, −4.70314455648781525373388489191, −3.91960856868223143826292320439, −3.64874573221102016790666356547, −3.31145774103875248086320466351, −2.28213109469766123290635461936, −1.56855577367282310331270726187, −0.877347881025579928283663922091, 0.877347881025579928283663922091, 1.56855577367282310331270726187, 2.28213109469766123290635461936, 3.31145774103875248086320466351, 3.64874573221102016790666356547, 3.91960856868223143826292320439, 4.70314455648781525373388489191, 5.42232139811458997298984942476, 5.65309758648045018186085625163, 6.31251491568464392148089374759, 6.57774176915863429748029454892, 6.99088542322115733938972134956, 7.71581291411964917884463964237, 8.188878433695120589679764983709, 8.580277831204113965788041851449

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