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Properties

Label 4-1769472-1.1-c1e2-0-18
Degree $4$
Conductor $1769472$
Sign $1$
Analytic cond. $112.823$
Root an. cond. $3.25911$
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 + 9-s + 4·11-s − 4·17-s + 12·19-s − 6·25-s + 27-s + 4·33-s − 12·41-s + 4·43-s − 6·49-s − 4·51-s + 12·57-s + 8·59-s − 12·73-s − 6·75-s + 81-s + 4·83-s + 24·89-s − 12·97-s + 4·99-s + 24·113-s − 6·121-s − 12·123-s + 127-s + 4·129-s + 131-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 1.20·11-s − 0.970·17-s + 2.75·19-s − 6/5·25-s + 0.192·27-s + 0.696·33-s − 1.87·41-s + 0.609·43-s − 6/7·49-s − 0.560·51-s + 1.58·57-s + 1.04·59-s − 1.40·73-s − 0.692·75-s + 1/9·81-s + 0.439·83-s + 2.54·89-s − 1.21·97-s + 0.402·99-s + 2.25·113-s − 0.545·121-s − 1.08·123-s + 0.0887·127-s + 0.352·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1769472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1769472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1769472\)    =    \(2^{16} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(112.823\)
Root analytic conductor: \(3.25911\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1769472,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.077534149\)
\(L(\frac12)\) \(\approx\) \(3.077534149\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.11.ae_w
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.e_w
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.19.am_cs
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.29.a_aby
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.37.a_abe
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.ae_cc
47$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.47.a_aco
53$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.53.a_be
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.ai_cs
61$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.61.a_adq
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.71.a_be
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.m_eo
79$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.79.a_cs
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.83.ae_cs
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.89.ay_la
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.m_ig
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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

−7.86702870582686351971112714988, −7.35049861374360485095925204446, −7.04452864015870064786456824612, −6.64582871229188279707731071299, −6.15209650774757189668687296107, −5.64856520770522339092245806644, −5.20186974142274193377185407761, −4.71455088705051495899696434294, −4.18991038285219330643504687256, −3.65201022226537207044264106927, −3.32376813133366639732036493331, −2.82156489045048743210148737326, −1.96295783561452911263517215517, −1.57233858610084808926333023767, −0.73207926381039395817779208925, 0.73207926381039395817779208925, 1.57233858610084808926333023767, 1.96295783561452911263517215517, 2.82156489045048743210148737326, 3.32376813133366639732036493331, 3.65201022226537207044264106927, 4.18991038285219330643504687256, 4.71455088705051495899696434294, 5.20186974142274193377185407761, 5.64856520770522339092245806644, 6.15209650774757189668687296107, 6.64582871229188279707731071299, 7.04452864015870064786456824612, 7.35049861374360485095925204446, 7.86702870582686351971112714988

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