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Properties

Label 4-93312-1.1-c1e2-0-19
Degree $4$
Conductor $93312$
Sign $1$
Analytic cond. $5.94965$
Root an. cond. $1.56179$
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  + 2-s + 4-s + 3·5-s + 8-s + 3·10-s + 16-s + 4·19-s + 3·20-s + 3·23-s − 25-s − 3·29-s + 32-s + 4·38-s + 3·40-s − 2·43-s + 3·46-s + 15·47-s − 4·49-s − 50-s − 9·53-s − 3·58-s + 64-s + 67-s − 9·71-s − 14·73-s + 4·76-s + 3·80-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.353·8-s + 0.948·10-s + 1/4·16-s + 0.917·19-s + 0.670·20-s + 0.625·23-s − 1/5·25-s − 0.557·29-s + 0.176·32-s + 0.648·38-s + 0.474·40-s − 0.304·43-s + 0.442·46-s + 2.18·47-s − 4/7·49-s − 0.141·50-s − 1.23·53-s − 0.393·58-s + 1/8·64-s + 0.122·67-s − 1.06·71-s − 1.63·73-s + 0.458·76-s + 0.335·80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(93312\)    =    \(2^{7} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(5.94965\)
Root analytic conductor: \(1.56179\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 93312,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.039883816\)
\(L(\frac12)\) \(\approx\) \(3.039883816\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3 \( 1 \)
good5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.5.ad_k
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.11.a_af
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \) 2.17.a_al
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.ad_ai
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.29.d_cg
31$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.31.a_ai
37$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.37.a_bi
41$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.a_b
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.c_g
47$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.47.ap_fs
53$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.j_cs
59$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.a_bl
61$C_2^2$ \( 1 - 104 T^{2} + p^{2} T^{4} \) 2.61.a_aea
67$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ab_ek
71$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.j_ge
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.o_ek
79$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.79.a_bo
83$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.83.a_adi
89$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.89.a_acn
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.c_ek
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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

−9.707678009785505293032446280294, −9.146825666475601982002236692060, −8.848543709878333731647077979935, −8.035015799304717455751275274756, −7.39680522038144146312052455032, −7.17632354082684406385884323612, −6.26694480303698990075616677282, −6.02332739575532603580210687486, −5.49838269001842769265702709456, −5.01113828880685326138992577887, −4.35092759043790437472236470691, −3.57087191222736082935298638676, −2.89762088885868424882568128390, −2.16312313271599019441163164389, −1.38282389508305668533759550136, 1.38282389508305668533759550136, 2.16312313271599019441163164389, 2.89762088885868424882568128390, 3.57087191222736082935298638676, 4.35092759043790437472236470691, 5.01113828880685326138992577887, 5.49838269001842769265702709456, 6.02332739575532603580210687486, 6.26694480303698990075616677282, 7.17632354082684406385884323612, 7.39680522038144146312052455032, 8.035015799304717455751275274756, 8.848543709878333731647077979935, 9.146825666475601982002236692060, 9.707678009785505293032446280294

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