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Properties

Label 4-1080000-1.1-c1e2-0-4
Degree $4$
Conductor $1080000$
Sign $1$
Analytic cond. $68.8617$
Root an. cond. $2.88067$
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 + 3·3-s − 4-s + 3·6-s − 3·8-s + 6·9-s − 4·11-s − 3·12-s − 16-s − 3·17-s + 6·18-s − 4·19-s − 4·22-s − 9·24-s + 9·27-s + 5·32-s − 12·33-s − 3·34-s − 6·36-s − 4·38-s + 6·41-s − 3·43-s + 4·44-s − 3·48-s + 6·49-s − 9·51-s + 9·54-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s − 1/2·4-s + 1.22·6-s − 1.06·8-s + 2·9-s − 1.20·11-s − 0.866·12-s − 1/4·16-s − 0.727·17-s + 1.41·18-s − 0.917·19-s − 0.852·22-s − 1.83·24-s + 1.73·27-s + 0.883·32-s − 2.08·33-s − 0.514·34-s − 36-s − 0.648·38-s + 0.937·41-s − 0.457·43-s + 0.603·44-s − 0.433·48-s + 6/7·49-s − 1.26·51-s + 1.22·54-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1080000\)    =    \(2^{6} \cdot 3^{3} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(68.8617\)
Root analytic conductor: \(2.88067\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1080000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.776629850\)
\(L(\frac12)\) \(\approx\) \(3.776629850\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
3$C_2$ \( 1 - p T + p T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.e_z
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.13.a_ab
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.17.d_g
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.e_bm
23$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \) 2.23.a_abp
29$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.29.a_ai
31$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.31.a_abq
37$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.37.a_r
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.ag_k
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.43.d_cg
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.47.a_acg
53$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.53.a_aw
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.59.as_hi
61$C_2^2$ \( 1 + 59 T^{2} + p^{2} T^{4} \) 2.61.a_ch
67$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.67.av_jk
71$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \) 2.71.a_dz
73$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.73.al_gi
79$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \) 2.79.a_ew
83$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.83.ac_gk
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.89.c_fa
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.ag_gw
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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

−8.100609065987188346510245888420, −7.894692628318564018906848816678, −7.25582303227646462347758408057, −6.79162347961404911683296644443, −6.42240125609168154076813217252, −5.70241184086122364263911726449, −5.27606694649753023640449201608, −4.80849264132663190901587265280, −4.28758531005647428201827339308, −3.83433942817620845102028772501, −3.53939234610410633424604216411, −2.76230675526556535116761131555, −2.42130317464945476027387773472, −2.01194530129340629997058028177, −0.69314375282985718466281352658, 0.69314375282985718466281352658, 2.01194530129340629997058028177, 2.42130317464945476027387773472, 2.76230675526556535116761131555, 3.53939234610410633424604216411, 3.83433942817620845102028772501, 4.28758531005647428201827339308, 4.80849264132663190901587265280, 5.27606694649753023640449201608, 5.70241184086122364263911726449, 6.42240125609168154076813217252, 6.79162347961404911683296644443, 7.25582303227646462347758408057, 7.894692628318564018906848816678, 8.100609065987188346510245888420

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