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Properties

Label 4-2100e2-1.1-c1e2-0-27
Degree $4$
Conductor $4410000$
Sign $1$
Analytic cond. $281.185$
Root an. cond. $4.09494$
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  − 3·3-s − 5·7-s + 6·9-s + 15·19-s + 15·21-s − 9·27-s − 18·31-s − 37-s − 16·43-s + 18·49-s − 45·57-s − 27·61-s − 30·63-s − 5·67-s − 27·73-s + 13·79-s + 9·81-s + 54·93-s + 33·103-s − 19·109-s + 3·111-s − 11·121-s + 127-s + 48·129-s + 131-s − 75·133-s + 137-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.88·7-s + 2·9-s + 3.44·19-s + 3.27·21-s − 1.73·27-s − 3.23·31-s − 0.164·37-s − 2.43·43-s + 18/7·49-s − 5.96·57-s − 3.45·61-s − 3.77·63-s − 0.610·67-s − 3.16·73-s + 1.46·79-s + 81-s + 5.59·93-s + 3.25·103-s − 1.81·109-s + 0.284·111-s − 121-s + 0.0887·127-s + 4.22·129-s + 0.0873·131-s − 6.50·133-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4410000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(281.185\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4410000,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.11.a_l
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.a_b
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) 2.19.ap_dq
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.23.a_x
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.s_fj
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.b_abk
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.53.a_cb
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 + 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.bb_ls
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.f_abq
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.73.bb_me
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.an_dm
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.89.a_adl
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.a_agl
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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

−8.959704987285887115583988433077, −8.944956716323593009340274014609, −7.75400839674116571239440330368, −7.59461213478708352559444032867, −7.15658939135890779930424894286, −7.08639148000656219215883809839, −6.38514587730014263435562691436, −6.16602466543640960225163255051, −5.62166634103015665937034679603, −5.56823798072336777183131269627, −4.86543766237316789326467093408, −4.83750551489742146452263981832, −3.78209517044340352980310984714, −3.52016939938474651814435791079, −3.25326601439805398984161234248, −2.62510793573990897062745453538, −1.47787717587236405967491373318, −1.28199931483748968222523711133, 0, 0, 1.28199931483748968222523711133, 1.47787717587236405967491373318, 2.62510793573990897062745453538, 3.25326601439805398984161234248, 3.52016939938474651814435791079, 3.78209517044340352980310984714, 4.83750551489742146452263981832, 4.86543766237316789326467093408, 5.56823798072336777183131269627, 5.62166634103015665937034679603, 6.16602466543640960225163255051, 6.38514587730014263435562691436, 7.08639148000656219215883809839, 7.15658939135890779930424894286, 7.59461213478708352559444032867, 7.75400839674116571239440330368, 8.944956716323593009340274014609, 8.959704987285887115583988433077

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