Properties

Label 4-342e2-1.1-c1e2-0-5
Degree $4$
Conductor $116964$
Sign $1$
Analytic cond. $7.45772$
Root an. cond. $1.65253$
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  + 4-s − 2·7-s + 16-s + 8·19-s + 2·25-s − 2·28-s + 4·43-s − 11·49-s + 16·61-s + 64-s + 22·73-s + 8·76-s + 2·100-s − 2·112-s − 10·121-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.755·7-s + 1/4·16-s + 1.83·19-s + 2/5·25-s − 0.377·28-s + 0.609·43-s − 1.57·49-s + 2.04·61-s + 1/8·64-s + 2.57·73-s + 0.917·76-s + 1/5·100-s − 0.188·112-s − 0.909·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116964 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116964 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(116964\)    =    \(2^{2} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(7.45772\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 116964,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.671046829\)
\(L(\frac12)\) \(\approx\) \(1.671046829\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.7.c_p
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.a_ax
17$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.17.a_bf
23$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.23.a_t
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.a_ax
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.43.ae_dm
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.47.a_de
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.a_z
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.59.a_ef
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \) 2.67.a_ach
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.73.aw_kh
79$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.79.a_aeg
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.83.a_cg
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.a_ac
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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

−9.605407065703912343416851780162, −9.099664708448324668905021423621, −8.432395833356932333111491246682, −7.917561078639665250780088901622, −7.50259449898580288804945304689, −6.82772899016339717101170695808, −6.63629908732757091523076348265, −5.93839100089462174116444512666, −5.35196356144638793869165345130, −4.95882909759922539371601914552, −4.02570731207988655165226312934, −3.38667134184301674532306741577, −2.94051047602386801244437832636, −2.08653099113308129105484111705, −0.957099345670396971730515506954, 0.957099345670396971730515506954, 2.08653099113308129105484111705, 2.94051047602386801244437832636, 3.38667134184301674532306741577, 4.02570731207988655165226312934, 4.95882909759922539371601914552, 5.35196356144638793869165345130, 5.93839100089462174116444512666, 6.63629908732757091523076348265, 6.82772899016339717101170695808, 7.50259449898580288804945304689, 7.917561078639665250780088901622, 8.432395833356932333111491246682, 9.099664708448324668905021423621, 9.605407065703912343416851780162

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