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Properties

Label 4-2646e2-1.1-c1e2-0-24
Degree $4$
Conductor $7001316$
Sign $1$
Analytic cond. $446.409$
Root an. cond. $4.59656$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 2·5-s + 4·8-s − 4·10-s + 2·11-s − 4·13-s + 5·16-s + 2·17-s + 4·19-s − 6·20-s + 4·22-s + 8·23-s − 8·26-s + 10·29-s − 4·31-s + 6·32-s + 4·34-s − 8·37-s + 8·38-s − 8·40-s + 6·41-s + 10·43-s + 6·44-s + 16·46-s − 6·47-s − 12·52-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s + 1.41·8-s − 1.26·10-s + 0.603·11-s − 1.10·13-s + 5/4·16-s + 0.485·17-s + 0.917·19-s − 1.34·20-s + 0.852·22-s + 1.66·23-s − 1.56·26-s + 1.85·29-s − 0.718·31-s + 1.06·32-s + 0.685·34-s − 1.31·37-s + 1.29·38-s − 1.26·40-s + 0.937·41-s + 1.52·43-s + 0.904·44-s + 2.35·46-s − 0.875·47-s − 1.66·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7001316\)    =    \(2^{2} \cdot 3^{6} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(446.409\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7001316,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.098903789\)
\(L(\frac12)\) \(\approx\) \(7.098903789\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_e
11$D_{4}$ \( 1 - 2 T + 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_q
13$D_{4}$ \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_x
17$D_{4}$ \( 1 - 2 T + 28 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_bc
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$D_{4}$ \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_bi
29$D_{4}$ \( 1 - 10 T + 76 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.29.ak_cy
31$D_{4}$ \( 1 + 4 T + 59 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_ch
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_bb
41$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_bc
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.43.ak_eh
47$D_{4}$ \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_bo
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$D_{4}$ \( 1 - 22 T + 232 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.59.aw_iy
61$D_{4}$ \( 1 - 20 T + 215 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.61.au_ih
67$D_{4}$ \( 1 - 6 T + 115 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_el
71$D_{4}$ \( 1 - 26 T + 304 T^{2} - 26 p T^{3} + p^{2} T^{4} \) 2.71.aba_ls
73$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.73.a_bi
79$D_{4}$ \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.79.e_an
83$D_{4}$ \( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.83.q_hu
89$D_{4}$ \( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_eu
97$D_{4}$ \( 1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_dn
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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

−8.800911124345058176243621146240, −8.755145885986163709020496053805, −8.116574850442499993552062840304, −7.83167003027978762715563741260, −7.31912267798861694119669611148, −7.02882458394887368259012825011, −6.75193310332819392767837271458, −6.58745089550047259641725770613, −5.57163083367294064065627507811, −5.55427470469490270124761544967, −5.10644225656728019526561751694, −4.83041716297234678835200908330, −4.15797439492400384816457250208, −3.89951477565617919270596375324, −3.55217522493453695792196316965, −3.05740726886877631625062705561, −2.36358404367415795768555100648, −2.36286529542835151260423417564, −1.11467340249133286939919662557, −0.813923008366618771072832583005, 0.813923008366618771072832583005, 1.11467340249133286939919662557, 2.36286529542835151260423417564, 2.36358404367415795768555100648, 3.05740726886877631625062705561, 3.55217522493453695792196316965, 3.89951477565617919270596375324, 4.15797439492400384816457250208, 4.83041716297234678835200908330, 5.10644225656728019526561751694, 5.55427470469490270124761544967, 5.57163083367294064065627507811, 6.58745089550047259641725770613, 6.75193310332819392767837271458, 7.02882458394887368259012825011, 7.31912267798861694119669611148, 7.83167003027978762715563741260, 8.116574850442499993552062840304, 8.755145885986163709020496053805, 8.800911124345058176243621146240

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