Properties

Degree $4$
Conductor $2480625$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s + 3·16-s − 49-s + 4·64-s − 4·79-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 2·4-s + 3·16-s − 49-s + 4·64-s − 4·79-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.976546573\)
\(L(\frac12)\) \(\approx\) \(1.976546573\)
\(L(1)\) 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)$
bad3 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good2$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_1$ \( ( 1 + T )^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{2} \)
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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

−9.777662703165187287566671656296, −9.751939668581427896467730196113, −8.785094445161926091883884528579, −8.696821163695744809947302107927, −8.120982520058901001726358376490, −7.69533806277893102285417128286, −7.25543411897011803892344680586, −7.20254116341639988099969785528, −6.42567958369940700772505248296, −6.41633951449973542666162314434, −5.72494990932537743023290213341, −5.62711434928389653058091435628, −4.85669050903742721122450969610, −4.44484030762349950054090385695, −3.53510049815431303241984854106, −3.47715774174445045269970651000, −2.61732237927340724392963698213, −2.49888726773389951467309249421, −1.66337124634595529707566199574, −1.27033692199437885483573765158, 1.27033692199437885483573765158, 1.66337124634595529707566199574, 2.49888726773389951467309249421, 2.61732237927340724392963698213, 3.47715774174445045269970651000, 3.53510049815431303241984854106, 4.44484030762349950054090385695, 4.85669050903742721122450969610, 5.62711434928389653058091435628, 5.72494990932537743023290213341, 6.41633951449973542666162314434, 6.42567958369940700772505248296, 7.20254116341639988099969785528, 7.25543411897011803892344680586, 7.69533806277893102285417128286, 8.120982520058901001726358376490, 8.696821163695744809947302107927, 8.785094445161926091883884528579, 9.751939668581427896467730196113, 9.777662703165187287566671656296

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