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Properties

Label 4-7440e2-1.1-c1e2-0-1
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s + 4·7-s + 3·9-s − 8·13-s + 4·15-s − 8·17-s − 8·21-s + 12·23-s + 3·25-s − 4·27-s − 4·29-s − 2·31-s − 8·35-s + 16·39-s + 4·41-s − 6·45-s + 4·47-s + 16·51-s − 8·53-s − 8·59-s − 4·61-s + 12·63-s + 16·65-s + 12·67-s − 24·69-s + 8·73-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 1.51·7-s + 9-s − 2.21·13-s + 1.03·15-s − 1.94·17-s − 1.74·21-s + 2.50·23-s + 3/5·25-s − 0.769·27-s − 0.742·29-s − 0.359·31-s − 1.35·35-s + 2.56·39-s + 0.624·41-s − 0.894·45-s + 0.583·47-s + 2.24·51-s − 1.09·53-s − 1.04·59-s − 0.512·61-s + 1.51·63-s + 1.98·65-s + 1.46·67-s − 2.88·69-s + 0.936·73-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: \(0\)
Selberg data: \((4,\ 55353600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.482114415\)
\(L(\frac12)\) \(\approx\) \(1.482114415\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_q
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.13.i_bo
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.17.i_by
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.19.a_be
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$D_{4}$ \( 1 + 4 T + 44 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_bs
37$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \) 2.37.a_cu
41$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_da
43$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.43.a_abq
47$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_dm
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.53.i_es
59$D_{4}$ \( 1 + 8 T + 116 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_em
61$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_cc
67$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.67.am_cu
71$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.71.a_bs
73$D_{4}$ \( 1 - 8 T + 64 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_cm
79$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_es
83$D_{4}$ \( 1 - 12 T + 170 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_go
89$D_{4}$ \( 1 + 4 T + 132 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.89.e_fc
97$D_{4}$ \( 1 - 20 T + 262 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.97.au_kc
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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

−7.85109625383074091524530253886, −7.67909629624958015939584541888, −7.23570002974757639498738593370, −7.21490170178832130939205784765, −6.70065373726535365261827172518, −6.48579704021006542156809930121, −5.94023812137233445137466139151, −5.50744895421640275081135967646, −4.94871140419464730161662075171, −4.78770510347286313383082424829, −4.72883425235970392775767471500, −4.59267606434569748800514349621, −3.66624845190436931284436502811, −3.62945974178702252387346971145, −2.68694936538832507762125830351, −2.56143809384325590381760667340, −1.80753270361967274804468633978, −1.68981471563679820153492271933, −0.59018378522351730703256707559, −0.54961145242667611209704256887, 0.54961145242667611209704256887, 0.59018378522351730703256707559, 1.68981471563679820153492271933, 1.80753270361967274804468633978, 2.56143809384325590381760667340, 2.68694936538832507762125830351, 3.62945974178702252387346971145, 3.66624845190436931284436502811, 4.59267606434569748800514349621, 4.72883425235970392775767471500, 4.78770510347286313383082424829, 4.94871140419464730161662075171, 5.50744895421640275081135967646, 5.94023812137233445137466139151, 6.48579704021006542156809930121, 6.70065373726535365261827172518, 7.21490170178832130939205784765, 7.23570002974757639498738593370, 7.67909629624958015939584541888, 7.85109625383074091524530253886

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