Properties

Label 4-1575e2-1.1-c1e2-0-8
Degree $4$
Conductor $2480625$
Sign $1$
Analytic cond. $158.166$
Root an. cond. $3.54632$
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  + 3·4-s + 5·16-s + 16·19-s − 4·29-s + 8·31-s + 12·41-s − 49-s + 8·59-s − 4·61-s + 3·64-s + 24·71-s + 48·76-s − 16·79-s − 12·89-s + 20·101-s + 36·109-s − 12·116-s − 22·121-s + 24·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 3/2·4-s + 5/4·16-s + 3.67·19-s − 0.742·29-s + 1.43·31-s + 1.87·41-s − 1/7·49-s + 1.04·59-s − 0.512·61-s + 3/8·64-s + 2.84·71-s + 5.50·76-s − 1.80·79-s − 1.27·89-s + 1.99·101-s + 3.44·109-s − 1.11·116-s − 2·121-s + 2.15·124-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2480625\)    =    \(3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(158.166\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2480625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.584011520\)
\(L(\frac12)\) \(\approx\) \(4.584011520\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−9.515463804908827023448684501502, −9.490983113544726911096892281442, −8.894731256731710977761461520416, −8.232235082359792221382278554871, −7.942708496971895126372494898041, −7.48858141921298755786638729116, −7.15692283948592883349122970358, −7.10458527971566872573636769625, −6.26943173189725074943512818816, −6.10638354132874879162814788058, −5.54589314464760391558465856111, −5.24483514627028430888491547916, −4.74574884356674026826544923732, −4.06139191848469787851416503012, −3.36565902875426010678441343639, −3.21345964266867327788660196480, −2.55451499972035659248183105405, −2.20307336360287510509149123793, −1.23092617630934680890913793523, −0.975673123889942811275946981400, 0.975673123889942811275946981400, 1.23092617630934680890913793523, 2.20307336360287510509149123793, 2.55451499972035659248183105405, 3.21345964266867327788660196480, 3.36565902875426010678441343639, 4.06139191848469787851416503012, 4.74574884356674026826544923732, 5.24483514627028430888491547916, 5.54589314464760391558465856111, 6.10638354132874879162814788058, 6.26943173189725074943512818816, 7.10458527971566872573636769625, 7.15692283948592883349122970358, 7.48858141921298755786638729116, 7.942708496971895126372494898041, 8.232235082359792221382278554871, 8.894731256731710977761461520416, 9.490983113544726911096892281442, 9.515463804908827023448684501502

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