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Properties

Label 4-740772-1.1-c1e2-0-3
Degree $4$
Conductor $740772$
Sign $1$
Analytic cond. $47.2322$
Root an. cond. $2.62155$
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  − 3-s + 4-s + 9-s − 12-s + 16-s + 19-s + 25-s − 27-s + 17·29-s + 36-s + 41-s − 9·43-s − 48-s − 13·49-s + 15·53-s − 57-s − 7·59-s − 2·61-s + 64-s + 14·71-s − 8·73-s − 75-s + 76-s + 81-s − 17·87-s − 2·89-s + 100-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s + 1/3·9-s − 0.288·12-s + 1/4·16-s + 0.229·19-s + 1/5·25-s − 0.192·27-s + 3.15·29-s + 1/6·36-s + 0.156·41-s − 1.37·43-s − 0.144·48-s − 1.85·49-s + 2.06·53-s − 0.132·57-s − 0.911·59-s − 0.256·61-s + 1/8·64-s + 1.66·71-s − 0.936·73-s − 0.115·75-s + 0.114·76-s + 1/9·81-s − 1.82·87-s − 0.211·89-s + 1/10·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(740772\)    =    \(2^{2} \cdot 3^{3} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(47.2322\)
Root analytic conductor: \(2.62155\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 740772,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.891491668\)
\(L(\frac12)\) \(\approx\) \(1.891491668\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 + T \)
19$C_1$ \( 1 - T \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.11.a_ab
13$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.13.a_t
17$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.17.a_g
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$ \( ( 1 - 9 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.29.ar_fa
31$C_2^2$ \( 1 - 36 T^{2} + p^{2} T^{4} \) 2.31.a_abk
37$C_2^2$ \( 1 + 39 T^{2} + p^{2} T^{4} \) 2.37.a_bn
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.ab_ai
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.j_dq
47$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \) 2.47.a_y
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.53.ap_ge
59$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.h_eu
61$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.c_es
67$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.67.a_f
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.71.ao_gk
73$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.73.i_fq
79$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.79.a_cw
83$C_2^2$ \( 1 + 105 T^{2} + p^{2} T^{4} \) 2.83.a_eb
89$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.c_fy
97$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.97.a_au
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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

−8.152025660908239030205934099555, −7.990015172186437913392116207288, −7.25238855970132591132487020796, −6.80786445563556260856664582317, −6.57895209564735994948151257858, −6.13883804438330140993212515652, −5.58081648047654174908596658332, −5.05916911691596608199504186861, −4.64516229073285011960678864689, −4.20975423508657598750437296185, −3.33222945265758899268895072413, −3.00515423082943548256771081130, −2.28705329279368846784334194802, −1.50224245585708132070581986573, −0.73186612623875233427404397957, 0.73186612623875233427404397957, 1.50224245585708132070581986573, 2.28705329279368846784334194802, 3.00515423082943548256771081130, 3.33222945265758899268895072413, 4.20975423508657598750437296185, 4.64516229073285011960678864689, 5.05916911691596608199504186861, 5.58081648047654174908596658332, 6.13883804438330140993212515652, 6.57895209564735994948151257858, 6.80786445563556260856664582317, 7.25238855970132591132487020796, 7.990015172186437913392116207288, 8.152025660908239030205934099555

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