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Properties

Label 4-144000-1.1-c1e2-0-9
Degree $4$
Conductor $144000$
Sign $1$
Analytic cond. $9.18156$
Root an. cond. $1.74072$
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 + 4·13-s + 16-s − 18-s + 20-s + 25-s − 4·26-s + 4·31-s − 32-s + 36-s − 8·37-s − 40-s + 16·41-s − 12·43-s + 45-s − 2·49-s − 50-s + 4·52-s + 8·53-s − 4·62-s + 64-s + 4·65-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 + 1.10·13-s + 1/4·16-s − 0.235·18-s + 0.223·20-s + 1/5·25-s − 0.784·26-s + 0.718·31-s − 0.176·32-s + 1/6·36-s − 1.31·37-s − 0.158·40-s + 2.49·41-s − 1.82·43-s + 0.149·45-s − 2/7·49-s − 0.141·50-s + 0.554·52-s + 1.09·53-s − 0.508·62-s + 1/8·64-s + 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(144000\)    =    \(2^{7} \cdot 3^{2} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(9.18156\)
Root analytic conductor: \(1.74072\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 144000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.381692069\)
\(L(\frac12)\) \(\approx\) \(1.381692069\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( 1 - T \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.13.ae_ba
17$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.17.a_g
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.23.a_aba
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.31.ae_ck
37$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.i_cw
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.41.aq_fa
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.43.m_es
47$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.47.a_w
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.ai_di
59$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.59.a_cw
61$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.61.a_g
67$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.67.ai_fq
71$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.71.am_da
73$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \) 2.73.a_aes
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.q_ig
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.89.aq_hy
97$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.97.a_abi
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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

−9.239017929550746735404791574089, −8.910091893541353432384639325958, −8.325019523634666183898891526733, −8.061063584539329861794145068359, −7.37114174944569447305046653449, −6.85809946022651850771745167896, −6.44086511283195212789530640624, −5.92907236132639256710655793569, −5.38165915968840011544461761118, −4.74446027620911328840173259317, −3.96007708504890879416393608699, −3.42217104958819547032166109942, −2.58654312151044494480774162871, −1.82377880686306950663111542047, −0.955895064069410118445920385164, 0.955895064069410118445920385164, 1.82377880686306950663111542047, 2.58654312151044494480774162871, 3.42217104958819547032166109942, 3.96007708504890879416393608699, 4.74446027620911328840173259317, 5.38165915968840011544461761118, 5.92907236132639256710655793569, 6.44086511283195212789530640624, 6.85809946022651850771745167896, 7.37114174944569447305046653449, 8.061063584539329861794145068359, 8.325019523634666183898891526733, 8.910091893541353432384639325958, 9.239017929550746735404791574089

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