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Properties

Label 4-234e2-1.1-c1e2-0-19
Degree $4$
Conductor $54756$
Sign $-1$
Analytic cond. $3.49129$
Root an. cond. $1.36693$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4-s + 9-s + 12-s − 2·13-s + 16-s − 3·17-s − 6·23-s − 2·25-s − 27-s − 3·29-s − 36-s + 2·39-s − 5·43-s − 48-s + 8·49-s + 3·51-s + 2·52-s − 6·53-s − 10·61-s − 64-s + 3·68-s + 6·69-s + 2·75-s − 79-s + 81-s + 3·87-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s + 1/3·9-s + 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.727·17-s − 1.25·23-s − 2/5·25-s − 0.192·27-s − 0.557·29-s − 1/6·36-s + 0.320·39-s − 0.762·43-s − 0.144·48-s + 8/7·49-s + 0.420·51-s + 0.277·52-s − 0.824·53-s − 1.28·61-s − 1/8·64-s + 0.363·68-s + 0.722·69-s + 0.230·75-s − 0.112·79-s + 1/9·81-s + 0.321·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(54756\)    =    \(2^{2} \cdot 3^{4} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(3.49129\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 54756,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( 1 + T \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.7.a_ai
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.11.a_f
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.23.g_cd
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.29.d_cg
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.37.a_bx
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.41.a_ba
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.f_dm
47$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.47.a_az
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.g_bi
59$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \) 2.59.a_bv
61$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.61.k_eh
67$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \) 2.67.a_dw
71$C_2^2$ \( 1 - 79 T^{2} + p^{2} T^{4} \) 2.71.a_adb
73$C_2^2$ \( 1 + 115 T^{2} + p^{2} T^{4} \) 2.73.a_el
79$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.79.b_ga
83$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.83.a_abi
89$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.89.a_aeo
97$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.97.a_acw
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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

−9.732723075867602610316552898217, −9.380712241449866121832833468113, −8.838548192591399037620664740553, −8.135281763544972474576128083420, −7.80992767234324173957377645101, −7.09933957405362551765252630149, −6.61457845223479812896775399852, −5.92740117553373818315460534007, −5.50829466773508155205121917228, −4.79205629351399060421149541391, −4.26508422810883323190323650843, −3.67616728917512678306703458653, −2.63269193819256613718370323625, −1.67236206976477827737398133156, 0, 1.67236206976477827737398133156, 2.63269193819256613718370323625, 3.67616728917512678306703458653, 4.26508422810883323190323650843, 4.79205629351399060421149541391, 5.50829466773508155205121917228, 5.92740117553373818315460534007, 6.61457845223479812896775399852, 7.09933957405362551765252630149, 7.80992767234324173957377645101, 8.135281763544972474576128083420, 8.838548192591399037620664740553, 9.380712241449866121832833468113, 9.732723075867602610316552898217

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