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Properties

Label 4-700e2-1.1-c1e2-0-8
Degree $4$
Conductor $490000$
Sign $1$
Analytic cond. $31.2428$
Root an. cond. $2.36421$
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  + 3-s + 4·7-s + 3·9-s + 3·11-s − 4·13-s + 3·17-s + 19-s + 4·21-s + 3·23-s + 8·27-s − 12·29-s + 7·31-s + 3·33-s − 37-s − 4·39-s + 12·41-s + 8·43-s − 9·47-s + 9·49-s + 3·51-s + 3·53-s + 57-s − 9·59-s + 61-s + 12·63-s − 7·67-s + 3·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s + 9-s + 0.904·11-s − 1.10·13-s + 0.727·17-s + 0.229·19-s + 0.872·21-s + 0.625·23-s + 1.53·27-s − 2.22·29-s + 1.25·31-s + 0.522·33-s − 0.164·37-s − 0.640·39-s + 1.87·41-s + 1.21·43-s − 1.31·47-s + 9/7·49-s + 0.420·51-s + 0.412·53-s + 0.132·57-s − 1.17·59-s + 0.128·61-s + 1.51·63-s − 0.855·67-s + 0.361·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(490000\)    =    \(2^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.2428\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 490000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.438891173\)
\(L(\frac12)\) \(\approx\) \(3.438891173\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_ac
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.ab_as
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_ao
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.ah_s
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.b_abk
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.47.j_bi
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_abs
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.59.j_w
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.ab_aci
67$C_2^2$ \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.67.h_as
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \) 2.73.b_acu
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.an_dm
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2^2$ \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.89.p_fg
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−10.48901972068012868972455417702, −10.35508363992139003417916379326, −9.563724693824625332199937660980, −9.458114382926935858904750426256, −9.045154973994390634652297060305, −8.427759257669920933694000067125, −8.068373590334878660937031328481, −7.59618411399666916661370408777, −7.15087059010579632517495786401, −7.12794565626510936920685402125, −6.10186939289148981230781709907, −5.81820520540124750831627349732, −4.92643690093785376021988999200, −4.82575660321742850830212789460, −4.21252357604103652490734770425, −3.76672952584091699620379785258, −2.94732356143077770675647655073, −2.38062798038959765339881574384, −1.59219404940058172339480430264, −1.10346059572545146215595998039, 1.10346059572545146215595998039, 1.59219404940058172339480430264, 2.38062798038959765339881574384, 2.94732356143077770675647655073, 3.76672952584091699620379785258, 4.21252357604103652490734770425, 4.82575660321742850830212789460, 4.92643690093785376021988999200, 5.81820520540124750831627349732, 6.10186939289148981230781709907, 7.12794565626510936920685402125, 7.15087059010579632517495786401, 7.59618411399666916661370408777, 8.068373590334878660937031328481, 8.427759257669920933694000067125, 9.045154973994390634652297060305, 9.458114382926935858904750426256, 9.563724693824625332199937660980, 10.35508363992139003417916379326, 10.48901972068012868972455417702

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