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Properties

Label 4-416000-1.1-c1e2-0-7
Degree $4$
Conductor $416000$
Sign $1$
Analytic cond. $26.5245$
Root an. cond. $2.26940$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 3·9-s + 13-s + 15·17-s + 25-s − 29-s + 15·37-s − 3·41-s + 3·45-s − 5·49-s + 18·53-s − 20·61-s − 65-s − 6·73-s − 15·85-s − 6·89-s − 6·97-s − 3·101-s − 10·109-s + 3·113-s − 3·117-s + 21·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.447·5-s − 9-s + 0.277·13-s + 3.63·17-s + 1/5·25-s − 0.185·29-s + 2.46·37-s − 0.468·41-s + 0.447·45-s − 5/7·49-s + 2.47·53-s − 2.56·61-s − 0.124·65-s − 0.702·73-s − 1.62·85-s − 0.635·89-s − 0.609·97-s − 0.298·101-s − 0.957·109-s + 0.282·113-s − 0.277·117-s + 1.90·121-s − 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(416000\)    =    \(2^{8} \cdot 5^{3} \cdot 13\)
Sign: $1$
Analytic conductor: \(26.5245\)
Root analytic conductor: \(2.26940\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 416000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.751417569\)
\(L(\frac12)\) \(\approx\) \(1.751417569\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.3.a_d
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \) 2.11.a_av
17$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) 2.17.ap_dm
19$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.19.a_b
23$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \) 2.23.a_y
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.29.b_bm
31$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.31.a_w
37$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.37.ap_ey
41$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.d_cm
43$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.43.a_bj
47$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \) 2.47.a_abh
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.53.as_hf
59$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.59.a_ad
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.67.a_du
71$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.71.a_au
73$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.g_fq
79$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \) 2.79.a_dr
83$C_2^2$ \( 1 - 48 T^{2} + p^{2} T^{4} \) 2.83.a_abw
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.g_gw
97$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.97.g_hr
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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

−8.520825155044296510715173552944, −8.076250094897980229668175861505, −7.75473471210819113912660749436, −7.46947687196604437547051880157, −6.86363155570309419142996046438, −6.00945416172410881307975772858, −5.86476248828868460747489612058, −5.46780652490468165367815690561, −4.87251030697865181557533044697, −4.15509803936434498442397082694, −3.62488016201665112817252246972, −3.01058872425441889188508442091, −2.78602477170844425612452951575, −1.51816181078463610641028780413, −0.78651806460918246307009197005, 0.78651806460918246307009197005, 1.51816181078463610641028780413, 2.78602477170844425612452951575, 3.01058872425441889188508442091, 3.62488016201665112817252246972, 4.15509803936434498442397082694, 4.87251030697865181557533044697, 5.46780652490468165367815690561, 5.86476248828868460747489612058, 6.00945416172410881307975772858, 6.86363155570309419142996046438, 7.46947687196604437547051880157, 7.75473471210819113912660749436, 8.076250094897980229668175861505, 8.520825155044296510715173552944

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