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Properties

Label 4-24e4-1.1-c1e2-0-55
Degree $4$
Conductor $331776$
Sign $1$
Analytic cond. $21.1543$
Root an. cond. $2.14461$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 3·7-s − 3·9-s − 5·11-s − 5·13-s − 4·17-s − 8·19-s − 23-s + 5·25-s − 9·29-s − 31-s + 3·35-s + 12·37-s − 3·41-s − 43-s + 3·45-s − 3·47-s + 7·49-s − 4·53-s + 5·55-s − 11·59-s + 7·61-s + 9·63-s + 5·65-s + 67-s − 8·71-s − 4·73-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.13·7-s − 9-s − 1.50·11-s − 1.38·13-s − 0.970·17-s − 1.83·19-s − 0.208·23-s + 25-s − 1.67·29-s − 0.179·31-s + 0.507·35-s + 1.97·37-s − 0.468·41-s − 0.152·43-s + 0.447·45-s − 0.437·47-s + 49-s − 0.549·53-s + 0.674·55-s − 1.43·59-s + 0.896·61-s + 1.13·63-s + 0.620·65-s + 0.122·67-s − 0.949·71-s − 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(331776\)    =    \(2^{12} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(21.1543\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 331776,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ae
7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_c
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.11.f_o
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.f_m
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) 2.23.b_aw
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.29.j_ca
31$C_2^2$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) 2.31.b_abe
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_abg
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_abq
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.47.d_abm
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$C_2^2$ \( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.59.l_ck
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.61.ah_am
67$C_2^2$ \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \) 2.67.ab_aco
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.71.i_gc
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_ada
83$C_2^2$ \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \) 2.83.b_ade
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \) 2.89.bk_ti
97$C_2^2$ \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.97.an_cu
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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

−10.43492694304869057440819661767, −10.16818066743046897339602178808, −9.625237058287413707032893783758, −9.170502737496770101100013179605, −8.785961773662585763219100354660, −8.310401263809879282965846006096, −7.75047630207729020276856840071, −7.54395960506447038417362647672, −6.71086566929469028540821028833, −6.62603027517487974904034472353, −5.76532420064636348290992978902, −5.61528410610409978086240987530, −4.73935065587436680190402379325, −4.48240781200315362118044152200, −3.74771966390283986014329240191, −2.93162622077891326075837830644, −2.64620976574182958188337782782, −2.07782862536966958638799869133, 0, 0, 2.07782862536966958638799869133, 2.64620976574182958188337782782, 2.93162622077891326075837830644, 3.74771966390283986014329240191, 4.48240781200315362118044152200, 4.73935065587436680190402379325, 5.61528410610409978086240987530, 5.76532420064636348290992978902, 6.62603027517487974904034472353, 6.71086566929469028540821028833, 7.54395960506447038417362647672, 7.75047630207729020276856840071, 8.310401263809879282965846006096, 8.785961773662585763219100354660, 9.170502737496770101100013179605, 9.625237058287413707032893783758, 10.16818066743046897339602178808, 10.43492694304869057440819661767

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