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Properties

Label 4-9680e2-1.1-c1e2-0-11
Degree $4$
Conductor $93702400$
Sign $1$
Analytic cond. $5974.54$
Root an. cond. $8.79176$
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  − 2·3-s + 2·5-s − 2·7-s − 9-s + 4·13-s − 4·15-s − 4·19-s + 4·21-s + 4·23-s + 3·25-s + 6·27-s + 4·29-s − 4·35-s − 8·37-s − 8·39-s − 2·41-s − 10·43-s − 2·45-s − 18·47-s − 9·49-s − 4·53-s + 8·57-s + 2·61-s + 2·63-s + 8·65-s + 10·67-s − 8·69-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 0.755·7-s − 1/3·9-s + 1.10·13-s − 1.03·15-s − 0.917·19-s + 0.872·21-s + 0.834·23-s + 3/5·25-s + 1.15·27-s + 0.742·29-s − 0.676·35-s − 1.31·37-s − 1.28·39-s − 0.312·41-s − 1.52·43-s − 0.298·45-s − 2.62·47-s − 9/7·49-s − 0.549·53-s + 1.05·57-s + 0.256·61-s + 0.251·63-s + 0.992·65-s + 1.22·67-s − 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(93702400\)    =    \(2^{8} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(5974.54\)
Root analytic conductor: \(8.79176\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 93702400,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good3$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_f
7$D_{4}$ \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_n
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$D_{4}$ \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_bi
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_s
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.31.a_be
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$D_{4}$ \( 1 + 2 T + 11 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.41.c_l
43$D_{4}$ \( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_ef
47$D_{4}$ \( 1 + 18 T + 173 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.47.s_gr
53$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.53.e_da
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.59.a_ak
61$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.61.ac_et
67$D_{4}$ \( 1 - 10 T + 157 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.67.ak_gb
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.71.ai_gc
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.73.ai_gg
79$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_es
83$D_{4}$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_bq
89$D_{4}$ \( 1 + 6 T + 155 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_fz
97$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.97.m_dy
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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

−7.29730307113811949948465865313, −6.96056480412414674889323104537, −6.58866008155836280309450883840, −6.35503705140176780594227722634, −6.22651709729512000016468343765, −6.07267795248099846897425197871, −5.30721348410075514974921456278, −5.19215533668999173471710283176, −4.87692963999750943163856434019, −4.74539390599279573730079697576, −3.79869626696907905087233393187, −3.65809382980077643966243739431, −3.17044978635934989500553666789, −3.00553733459205509686764162969, −2.22281723975367211310787491714, −2.00293719542701549633808443107, −1.32895528378198871942720540511, −0.993201180404461114608620942651, 0, 0, 0.993201180404461114608620942651, 1.32895528378198871942720540511, 2.00293719542701549633808443107, 2.22281723975367211310787491714, 3.00553733459205509686764162969, 3.17044978635934989500553666789, 3.65809382980077643966243739431, 3.79869626696907905087233393187, 4.74539390599279573730079697576, 4.87692963999750943163856434019, 5.19215533668999173471710283176, 5.30721348410075514974921456278, 6.07267795248099846897425197871, 6.22651709729512000016468343765, 6.35503705140176780594227722634, 6.58866008155836280309450883840, 6.96056480412414674889323104537, 7.29730307113811949948465865313

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