↑

Properties

Label 4-1037232-1.1-c1e2-0-2
Degree $4$
Conductor $1037232$
Sign $1$
Analytic cond. $66.1348$
Root an. cond. $2.85172$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s + 6·13-s + 2·19-s − 6·25-s + 27-s − 6·31-s − 2·37-s + 6·39-s + 22·43-s + 2·57-s + 12·61-s + 26·67-s + 22·73-s − 6·75-s − 6·79-s + 81-s − 6·93-s − 20·97-s − 22·103-s − 22·109-s − 2·111-s + 6·117-s − 18·121-s + 127-s + 22·129-s + 131-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 1.66·13-s + 0.458·19-s − 6/5·25-s + 0.192·27-s − 1.07·31-s − 0.328·37-s + 0.960·39-s + 3.35·43-s + 0.264·57-s + 1.53·61-s + 3.17·67-s + 2.57·73-s − 0.692·75-s − 0.675·79-s + 1/9·81-s − 0.622·93-s − 2.03·97-s − 2.16·103-s − 2.10·109-s − 0.189·111-s + 0.554·117-s − 1.63·121-s + 0.0887·127-s + 1.93·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1037232 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1037232 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1037232\)    =    \(2^{4} \cdot 3^{3} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(66.1348\)
Root analytic conductor: \(2.85172\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1037232,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.955528550\)
\(L(\frac12)\) \(\approx\) \(2.955528550\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( 1 - T \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.13.ag_bj
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.19.ac_bn
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.31.g_ct
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.37.c_cx
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.43.aw_hz
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.a_abm
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.67.aba_lr
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.a_bq
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.73.aw_kh
79$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.79.g_gl
83$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.83.a_gg
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.213177915144848467457073975549, −7.86003472307450153648690839574, −7.14892757864943782004910226592, −6.92104634189874182322622514416, −6.35041177064936179659786450497, −5.84853197538479925165694215764, −5.44492897197805836950949219523, −5.10264611010005387817887955500, −3.98667962587873281431655095458, −3.97239648734814585129569787000, −3.65955604277278655695987739777, −2.72076192155102032854084811383, −2.31898745928895941643396981279, −1.52796226243779435664402358200, −0.814330480313867343133277258395, 0.814330480313867343133277258395, 1.52796226243779435664402358200, 2.31898745928895941643396981279, 2.72076192155102032854084811383, 3.65955604277278655695987739777, 3.97239648734814585129569787000, 3.98667962587873281431655095458, 5.10264611010005387817887955500, 5.44492897197805836950949219523, 5.84853197538479925165694215764, 6.35041177064936179659786450497, 6.92104634189874182322622514416, 7.14892757864943782004910226592, 7.86003472307450153648690839574, 8.213177915144848467457073975549

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