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Properties

Label 4-2700e2-1.1-c1e2-0-9
Degree $4$
Conductor $7290000$
Sign $1$
Analytic cond. $464.816$
Root an. cond. $4.64323$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·19-s + 12·29-s + 10·31-s + 24·41-s + 10·49-s − 12·59-s − 14·61-s + 24·71-s + 2·79-s + 24·89-s + 24·101-s + 14·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 2.29·19-s + 2.22·29-s + 1.79·31-s + 3.74·41-s + 10/7·49-s − 1.56·59-s − 1.79·61-s + 2.84·71-s + 0.225·79-s + 2.54·89-s + 2.38·101-s + 1.34·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7290000\)    =    \(2^{4} \cdot 3^{6} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(464.816\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7290000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.788461138\)
\(L(\frac12)\) \(\approx\) \(2.788461138\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.17.a_az
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.19.k_cl
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.23.a_abl
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.31.ak_dj
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.41.ay_is
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \) 2.53.a_adt
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.59.m_fy
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.61.o_gp
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.79.ac_gd
83$C_2^2$ \( 1 + 59 T^{2} + p^{2} T^{4} \) 2.83.a_ch
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.89.ay_mk
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.97.a_ck
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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

−8.961674238755181288555351983914, −8.641376715022399568844818577490, −8.250100180163382364451055190830, −7.988040610800715474997577426807, −7.53765535799600435331485623999, −7.22263559902706455664882832677, −6.45021323580409776642391681038, −6.38043315588370698603746894730, −6.15580533544853977406048423001, −5.71924210767766422924781796538, −4.89264311052844638024133501894, −4.70153302287344898587962041331, −4.28316292310072506547373353798, −4.07376831333639093847210195477, −3.26878408038463140429288213741, −2.86139760477548922546611804063, −2.24121250919515878417891747214, −2.14309674741299694003115224713, −0.980092602062828765124024702493, −0.67521830249622127485093507196, 0.67521830249622127485093507196, 0.980092602062828765124024702493, 2.14309674741299694003115224713, 2.24121250919515878417891747214, 2.86139760477548922546611804063, 3.26878408038463140429288213741, 4.07376831333639093847210195477, 4.28316292310072506547373353798, 4.70153302287344898587962041331, 4.89264311052844638024133501894, 5.71924210767766422924781796538, 6.15580533544853977406048423001, 6.38043315588370698603746894730, 6.45021323580409776642391681038, 7.22263559902706455664882832677, 7.53765535799600435331485623999, 7.988040610800715474997577426807, 8.250100180163382364451055190830, 8.641376715022399568844818577490, 8.961674238755181288555351983914

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