Properties

Label 4-6075e2-1.1-c1e2-0-4
Degree $4$
Conductor $36905625$
Sign $1$
Analytic cond. $2353.13$
Root an. cond. $6.96484$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s − 4·7-s + 2·13-s − 2·19-s − 8·28-s − 2·31-s − 16·37-s − 22·43-s − 2·49-s + 4·52-s + 10·61-s − 8·64-s + 14·67-s − 22·73-s − 4·76-s − 14·79-s − 8·91-s + 14·97-s + 14·103-s − 2·109-s − 16·121-s − 4·124-s + 127-s + 131-s + 8·133-s + 137-s + 139-s + ⋯
L(s)  = 1  + 4-s − 1.51·7-s + 0.554·13-s − 0.458·19-s − 1.51·28-s − 0.359·31-s − 2.63·37-s − 3.35·43-s − 2/7·49-s + 0.554·52-s + 1.28·61-s − 64-s + 1.71·67-s − 2.57·73-s − 0.458·76-s − 1.57·79-s − 0.838·91-s + 1.42·97-s + 1.37·103-s − 0.191·109-s − 1.45·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.693·133-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36905625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36905625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36905625\)    =    \(3^{10} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2353.13\)
Root analytic conductor: \(6.96484\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 36905625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 112 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 88 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−7.896023416387316924987775456958, −7.24002361353899934476797604702, −7.14530190394130024227769269298, −6.77199008323373875461092611203, −6.60336360365791201562163024207, −6.19893807282492399989648060617, −5.93803341200205461626794735251, −5.46819545625682359071677540787, −4.99508428817044183506351901971, −4.77178676416567706532410830168, −4.10519787553572305042732635609, −3.60675821842974492129722607826, −3.36479820308400257402539502932, −3.14245968042950154191006223597, −2.62051425614622438214464670419, −1.88686163342032147582417204350, −1.87145940004656778494320204254, −1.15587054095223283486260837620, 0, 0, 1.15587054095223283486260837620, 1.87145940004656778494320204254, 1.88686163342032147582417204350, 2.62051425614622438214464670419, 3.14245968042950154191006223597, 3.36479820308400257402539502932, 3.60675821842974492129722607826, 4.10519787553572305042732635609, 4.77178676416567706532410830168, 4.99508428817044183506351901971, 5.46819545625682359071677540787, 5.93803341200205461626794735251, 6.19893807282492399989648060617, 6.60336360365791201562163024207, 6.77199008323373875461092611203, 7.14530190394130024227769269298, 7.24002361353899934476797604702, 7.896023416387316924987775456958

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