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Properties

Label 4-7440e2-1.1-c1e2-0-5
Degree $4$
Conductor $55353600$
Sign $1$
Analytic cond. $3529.39$
Root an. cond. $7.70770$
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 + 3·9-s + 8·11-s − 2·13-s + 4·15-s − 8·17-s + 4·19-s + 4·21-s + 4·23-s + 3·25-s − 4·27-s − 10·29-s + 2·31-s − 16·33-s + 4·35-s − 6·37-s + 4·39-s − 4·41-s − 4·43-s − 6·45-s + 8·47-s − 8·49-s + 16·51-s − 16·55-s − 8·57-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 0.755·7-s + 9-s + 2.41·11-s − 0.554·13-s + 1.03·15-s − 1.94·17-s + 0.917·19-s + 0.872·21-s + 0.834·23-s + 3/5·25-s − 0.769·27-s − 1.85·29-s + 0.359·31-s − 2.78·33-s + 0.676·35-s − 0.986·37-s + 0.640·39-s − 0.624·41-s − 0.609·43-s − 0.894·45-s + 1.16·47-s − 8/7·49-s + 2.24·51-s − 2.15·55-s − 1.05·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55353600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55353600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(55353600\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(3529.39\)
Root analytic conductor: \(7.70770\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 55353600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
31$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_m
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.11.ai_bm
13$D_{4}$ \( 1 + 2 T + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_a
17$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bm
19$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_be
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.23.ae_by
29$D_{4}$ \( 1 + 10 T + 80 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.29.k_dc
37$D_{4}$ \( 1 + 6 T + 56 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_ce
41$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_bm
43$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_da
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.47.ai_eg
53$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \) 2.53.a_dq
59$D_{4}$ \( 1 - 2 T + 92 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.59.ac_do
61$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_be
67$D_{4}$ \( 1 + 6 T + 116 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.67.g_em
71$D_{4}$ \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_fs
73$D_{4}$ \( 1 + 2 T - 96 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_ads
79$D_{4}$ \( 1 - 4 T + 114 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_ek
83$D_{4}$ \( 1 - 4 T - 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_aw
89$D_{4}$ \( 1 - 18 T + 232 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.89.as_iy
97$C_2^2$ \( 1 + 182 T^{2} + p^{2} T^{4} \) 2.97.a_ha
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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

−7.35666610083462546926344242454, −7.35536972514131774460796959561, −6.83244111805615310843956459750, −6.78528541590432921594060472585, −6.33638104835496966058172892991, −6.26340187324967490580166259797, −5.51045713358818007026596283893, −5.33833515636101811640967121681, −4.83518973027026580557992994013, −4.54455891490098740951448661643, −4.08106792438554592985897663397, −3.87002928659764465649957659154, −3.35317254848373758745252390191, −3.28157564467793502650368407221, −2.29979353560246442651823869609, −2.03884459743042658551017782201, −1.25038429543431843733467570903, −1.03734506640482995753095136845, 0, 0, 1.03734506640482995753095136845, 1.25038429543431843733467570903, 2.03884459743042658551017782201, 2.29979353560246442651823869609, 3.28157564467793502650368407221, 3.35317254848373758745252390191, 3.87002928659764465649957659154, 4.08106792438554592985897663397, 4.54455891490098740951448661643, 4.83518973027026580557992994013, 5.33833515636101811640967121681, 5.51045713358818007026596283893, 6.26340187324967490580166259797, 6.33638104835496966058172892991, 6.78528541590432921594060472585, 6.83244111805615310843956459750, 7.35536972514131774460796959561, 7.35666610083462546926344242454

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