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Properties

Label 4-52e4-1.1-c1e2-0-13
Degree $4$
Conductor $7311616$
Sign $1$
Analytic cond. $466.194$
Root an. cond. $4.64667$
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  − 4·3-s + 6·9-s + 6·17-s − 12·23-s − 7·25-s + 4·27-s + 6·29-s + 16·43-s − 14·49-s − 24·51-s − 6·53-s + 2·61-s + 48·69-s + 28·75-s − 8·79-s − 37·81-s − 24·87-s + 6·101-s − 20·103-s − 12·107-s − 30·113-s − 22·121-s + 127-s − 64·129-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 2.30·3-s + 2·9-s + 1.45·17-s − 2.50·23-s − 7/5·25-s + 0.769·27-s + 1.11·29-s + 2.43·43-s − 2·49-s − 3.36·51-s − 0.824·53-s + 0.256·61-s + 5.77·69-s + 3.23·75-s − 0.900·79-s − 4.11·81-s − 2.57·87-s + 0.597·101-s − 1.97·103-s − 1.16·107-s − 2.82·113-s − 2·121-s + 0.0887·127-s − 5.63·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7311616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7311616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7311616\)    =    \(2^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(466.194\)
Root analytic conductor: \(4.64667\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 7311616,\ (\ :1/2, 1/2),\ 1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
13 \( 1 \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.3.e_k
5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.19.a_ba
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.29.ag_cp
31$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.31.a_by
37$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.37.a_ab
41$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.41.a_cd
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.47.a_de
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.53.g_el
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.59.a_cs
61$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.61.ac_et
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \) 2.67.a_es
71$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \) 2.71.a_fa
73$C_2^2$ \( 1 + 143 T^{2} + p^{2} T^{4} \) 2.73.a_fn
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.83.a_aba
89$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \) 2.89.a_fa
97$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \) 2.97.a_fq
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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

−8.444149866012287590143563345955, −8.186295070532041850783313074484, −7.75615633600513197569075321142, −7.67608746156786528000319141645, −6.88246594813852617077076305641, −6.54749239762301718342349358369, −6.17258627122264253555949926971, −5.97204716783874157697727826239, −5.52143351833961382087231445502, −5.45489930101086479074373346726, −4.76697046243785410228269343441, −4.53208108545159031486032679717, −3.80565570669197829521371891953, −3.72419038875928673461677860178, −2.70814242881485983864390207886, −2.48331749157434785964490431688, −1.41040221424437537093781946900, −1.16704529545775230507542854150, 0, 0, 1.16704529545775230507542854150, 1.41040221424437537093781946900, 2.48331749157434785964490431688, 2.70814242881485983864390207886, 3.72419038875928673461677860178, 3.80565570669197829521371891953, 4.53208108545159031486032679717, 4.76697046243785410228269343441, 5.45489930101086479074373346726, 5.52143351833961382087231445502, 5.97204716783874157697727826239, 6.17258627122264253555949926971, 6.54749239762301718342349358369, 6.88246594813852617077076305641, 7.67608746156786528000319141645, 7.75615633600513197569075321142, 8.186295070532041850783313074484, 8.444149866012287590143563345955

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