Properties

Label 4-52000-1.1-c1e2-0-0
Degree $4$
Conductor $52000$
Sign $1$
Analytic cond. $3.31556$
Root an. cond. $1.34939$
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  − 2-s + 4-s − 5-s − 8-s − 9-s + 10-s + 7·13-s + 16-s − 17-s + 18-s − 20-s + 25-s − 7·26-s − 29-s − 32-s + 34-s − 36-s + 9·37-s + 40-s − 5·41-s + 45-s − 11·49-s − 50-s + 7·52-s + 20·53-s + 58-s + 8·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s − 1/3·9-s + 0.316·10-s + 1.94·13-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.223·20-s + 1/5·25-s − 1.37·26-s − 0.185·29-s − 0.176·32-s + 0.171·34-s − 1/6·36-s + 1.47·37-s + 0.158·40-s − 0.780·41-s + 0.149·45-s − 1.57·49-s − 0.141·50-s + 0.970·52-s + 2.74·53-s + 0.131·58-s + 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(52000\)    =    \(2^{5} \cdot 5^{3} \cdot 13\)
Sign: $1$
Analytic conductor: \(3.31556\)
Root analytic conductor: \(1.34939\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 52000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9397025068\)
\(L(\frac12)\) \(\approx\) \(0.9397025068\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5$C_1$ \( 1 + T \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 6 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.7.a_l
11$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.11.a_ab
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.17.b_o
19$C_2^2$ \( 1 + 21 T^{2} + p^{2} T^{4} \) 2.19.a_v
23$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.23.a_au
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) 2.29.b_cg
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
37$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.aj_dk
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.41.f_q
43$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.43.a_b
47$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \) 2.47.a_cn
53$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) 2.53.au_hx
59$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.59.a_z
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.ai_dy
67$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.67.a_cc
71$C_2^2$ \( 1 - 84 T^{2} + p^{2} T^{4} \) 2.71.a_adg
73$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.73.ao_go
79$C_2^2$ \( 1 - 89 T^{2} + p^{2} T^{4} \) 2.79.a_adl
83$C_2^2$ \( 1 + 88 T^{2} + p^{2} T^{4} \) 2.83.a_dk
89$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.89.ag_he
97$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.97.e_gr
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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

−10.17314329293074216601774574777, −9.427900790310690651432114174774, −9.007725271502181072171978190866, −8.449301783038200631769980507617, −8.195340118956356740393687462828, −7.65816334540193360822274910572, −6.84599404164933384342324838299, −6.54192471433407382683929771058, −5.85023478071209069305684427618, −5.36662009164489740071630255975, −4.37577611420169921232838048408, −3.75683922436386686881893736689, −3.13404690266031888712715932738, −2.12653902907953296813501177171, −0.968093441742912077454432774383, 0.968093441742912077454432774383, 2.12653902907953296813501177171, 3.13404690266031888712715932738, 3.75683922436386686881893736689, 4.37577611420169921232838048408, 5.36662009164489740071630255975, 5.85023478071209069305684427618, 6.54192471433407382683929771058, 6.84599404164933384342324838299, 7.65816334540193360822274910572, 8.195340118956356740393687462828, 8.449301783038200631769980507617, 9.007725271502181072171978190866, 9.427900790310690651432114174774, 10.17314329293074216601774574777

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