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Properties

Label 4-1782272-1.1-c1e2-0-2
Degree $4$
Conductor $1782272$
Sign $1$
Analytic cond. $113.639$
Root an. cond. $3.26499$
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  + 6·3-s + 21·9-s + 12·11-s − 4·17-s − 2·19-s − 25-s + 54·27-s + 72·33-s − 2·41-s − 20·43-s − 5·49-s − 24·51-s − 12·57-s − 2·59-s + 4·67-s − 16·73-s − 6·75-s + 108·81-s − 20·83-s − 32·89-s − 8·97-s + 252·99-s + 30·107-s − 12·113-s + 86·121-s − 12·123-s + 127-s + ⋯
L(s)  = 1  + 3.46·3-s + 7·9-s + 3.61·11-s − 0.970·17-s − 0.458·19-s − 1/5·25-s + 10.3·27-s + 12.5·33-s − 0.312·41-s − 3.04·43-s − 5/7·49-s − 3.36·51-s − 1.58·57-s − 0.260·59-s + 0.488·67-s − 1.87·73-s − 0.692·75-s + 12·81-s − 2.19·83-s − 3.39·89-s − 0.812·97-s + 25.3·99-s + 2.90·107-s − 1.12·113-s + 7.81·121-s − 1.08·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1782272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1782272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1782272\)    =    \(2^{9} \cdot 59^{2}\)
Sign: $1$
Analytic conductor: \(113.639\)
Root analytic conductor: \(3.26499\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1782272,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(11.07305889\)
\(L(\frac12)\) \(\approx\) \(11.07305889\)
\(L(\frac{3}{2})\) 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 \( 1 \)
59$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )^{2} \) 2.3.ag_p
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.11.am_cg
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.29.a_cf
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.a_cg
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.a_cg
41$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.41.c_df
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.43.u_he
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.53.a_dd
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.71.a_ew
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.73.q_ic
79$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.79.a_bl
83$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.83.u_kg
89$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.89.bg_qs
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.97.i_ic
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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

−8.139942568435135876416009527981, −7.33975666397590421622348553433, −7.12104801080175016984583410739, −6.72136610741303542487376695263, −6.51534276590235620522788710540, −5.84476510638913664881514984210, −4.74756935412399231087377330703, −4.26553399618190477828297127438, −4.10286479084315987002382830785, −3.70847715428605487295086359244, −3.06493530437300139702880016153, −3.03150756258435225512959129754, −1.97635636450260983961790313324, −1.68489164110150245010533612157, −1.40590255916755611764878380648, 1.40590255916755611764878380648, 1.68489164110150245010533612157, 1.97635636450260983961790313324, 3.03150756258435225512959129754, 3.06493530437300139702880016153, 3.70847715428605487295086359244, 4.10286479084315987002382830785, 4.26553399618190477828297127438, 4.74756935412399231087377330703, 5.84476510638913664881514984210, 6.51534276590235620522788710540, 6.72136610741303542487376695263, 7.12104801080175016984583410739, 7.33975666397590421622348553433, 8.139942568435135876416009527981

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