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Properties

Label 4-7440e2-1.1-c1e2-0-7
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·5-s − 2·7-s + 3·9-s − 2·13-s − 4·15-s − 4·19-s − 4·21-s + 8·23-s + 3·25-s + 4·27-s − 10·29-s + 2·31-s + 4·35-s + 14·37-s − 4·39-s − 12·41-s − 4·43-s − 6·45-s − 6·49-s + 8·53-s − 8·57-s − 18·59-s − 16·61-s − 6·63-s + 4·65-s + 6·67-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s − 0.755·7-s + 9-s − 0.554·13-s − 1.03·15-s − 0.917·19-s − 0.872·21-s + 1.66·23-s + 3/5·25-s + 0.769·27-s − 1.85·29-s + 0.359·31-s + 0.676·35-s + 2.30·37-s − 0.640·39-s − 1.87·41-s − 0.609·43-s − 0.894·45-s − 6/7·49-s + 1.09·53-s − 1.05·57-s − 2.34·59-s − 2.04·61-s − 0.755·63-s + 0.496·65-s + 0.733·67-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$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_k
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_w
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_w
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.23.ai_ck
29$D_{4}$ \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.29.k_da
37$D_{4}$ \( 1 - 14 T + 118 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.37.ao_eo
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_cs
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.47.a_o
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_dy
59$D_{4}$ \( 1 + 18 T + 194 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.59.s_hm
61$C_4$ \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.61.q_gk
67$D_{4}$ \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_du
71$D_{4}$ \( 1 - 14 T + 186 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.71.ao_he
73$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_w
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.79.a_da
83$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.83.i_dy
89$D_{4}$ \( 1 + 10 T + 198 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.89.k_hq
97$D_{4}$ \( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_hi
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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

−7.76457027675638482940565312428, −7.51708990473669529663731105131, −6.96596350302585212797500050856, −6.89117683122809147729774235816, −6.32503515894003481708323944250, −6.29760774435978589645437068097, −5.49720938395164302329753284660, −5.21016241952073019509681971012, −4.74625329726741567894112251353, −4.48372358576698542009860440303, −3.87872969493734445842300610872, −3.85846781963599991861573616918, −3.17472685513978422951436564329, −3.07418611720367838147622209905, −2.53858544188524856856524432084, −2.27321450279558722165411698198, −1.41771763413494464145050963483, −1.24125579237863329761189077616, 0, 0, 1.24125579237863329761189077616, 1.41771763413494464145050963483, 2.27321450279558722165411698198, 2.53858544188524856856524432084, 3.07418611720367838147622209905, 3.17472685513978422951436564329, 3.85846781963599991861573616918, 3.87872969493734445842300610872, 4.48372358576698542009860440303, 4.74625329726741567894112251353, 5.21016241952073019509681971012, 5.49720938395164302329753284660, 6.29760774435978589645437068097, 6.32503515894003481708323944250, 6.89117683122809147729774235816, 6.96596350302585212797500050856, 7.51708990473669529663731105131, 7.76457027675638482940565312428

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