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Properties

Label 4-2624-1.1-c1e2-0-0
Degree $4$
Conductor $2624$
Sign $1$
Analytic cond. $0.167308$
Root an. cond. $0.639557$
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  − 2·9-s − 2·13-s − 10·25-s + 6·29-s + 4·37-s − 7·41-s + 2·49-s + 18·53-s − 8·61-s + 4·73-s − 5·81-s + 12·89-s + 4·97-s + 30·101-s − 14·109-s − 12·113-s + 4·117-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 2/3·9-s − 0.554·13-s − 2·25-s + 1.11·29-s + 0.657·37-s − 1.09·41-s + 2/7·49-s + 2.47·53-s − 1.02·61-s + 0.468·73-s − 5/9·81-s + 1.27·89-s + 0.406·97-s + 2.98·101-s − 1.34·109-s − 1.12·113-s + 0.369·117-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $1$
Analytic conductor: \(0.167308\)
Root analytic conductor: \(0.639557\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2624,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6718363961\)
\(L(\frac12)\) \(\approx\) \(0.6718363961\)
\(L(\frac{3}{2})\) 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 \( 1 \)
41$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 6 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.c_s
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.29.ag_cg
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.ae_bq
43$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.43.a_adi
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.53.as_gw
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.59.a_k
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.i_dy
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.67.a_k
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.ae_g
79$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.79.a_cs
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−13.09400100773783770548626724440, −12.17122437926391485901416794358, −11.84972258756862714429114283013, −11.34438180625797519231678092248, −10.39472726048858312881499070397, −10.03491127413469791102707134955, −9.239002662037055803972611007000, −8.573059343842824820442031240008, −7.896918153096677951553749047847, −7.22455768358167539416208474085, −6.28614545798265012698517485443, −5.62602027401331625110196694077, −4.70549695994879504073981637924, −3.65115374605360031599677123521, −2.39797776147134135959499756883, 2.39797776147134135959499756883, 3.65115374605360031599677123521, 4.70549695994879504073981637924, 5.62602027401331625110196694077, 6.28614545798265012698517485443, 7.22455768358167539416208474085, 7.896918153096677951553749047847, 8.573059343842824820442031240008, 9.239002662037055803972611007000, 10.03491127413469791102707134955, 10.39472726048858312881499070397, 11.34438180625797519231678092248, 11.84972258756862714429114283013, 12.17122437926391485901416794358, 13.09400100773783770548626724440

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