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Properties

Label 4-2178e2-1.1-c1e2-0-18
Degree $4$
Conductor $4743684$
Sign $-1$
Analytic cond. $302.461$
Root an. cond. $4.17030$
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 − 2·9-s − 12-s + 16-s − 9·23-s − 10·25-s + 5·27-s + 2·31-s − 2·36-s + 5·37-s − 9·47-s − 48-s + 13·49-s − 3·53-s + 9·59-s + 64-s − 7·67-s + 9·69-s + 15·71-s + 10·75-s + 81-s − 9·92-s − 2·93-s + 10·97-s − 10·100-s + 7·103-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s − 2/3·9-s − 0.288·12-s + 1/4·16-s − 1.87·23-s − 2·25-s + 0.962·27-s + 0.359·31-s − 1/3·36-s + 0.821·37-s − 1.31·47-s − 0.144·48-s + 13/7·49-s − 0.412·53-s + 1.17·59-s + 1/8·64-s − 0.855·67-s + 1.08·69-s + 1.78·71-s + 1.15·75-s + 1/9·81-s − 0.938·92-s − 0.207·93-s + 1.01·97-s − 100-s + 0.689·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4743684\)    =    \(2^{2} \cdot 3^{4} \cdot 11^{4}\)
Sign: $-1$
Analytic conductor: \(302.461\)
Root analytic conductor: \(4.17030\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 4743684,\ (\ :1/2, 1/2),\ -1)\)

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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 + T + p T^{2} \)
11 \( 1 \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.7.a_an
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.13.a_c
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.j_cm
29$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.29.a_abg
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.ac_as
37$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.af_ci
41$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.41.a_bc
43$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \) 2.43.a_bp
47$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.j_cg
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.d_ca
59$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.aj_eo
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.h_ew
71$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ap_fm
73$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.73.a_di
79$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.79.a_az
83$C_2^2$ \( 1 + 139 T^{2} + p^{2} T^{4} \) 2.83.a_fj
89$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.89.a_gn
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.ak_ic
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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

−7.14625534855191965904252856545, −6.49465019242782090984922319554, −6.34456479306805090859466531980, −5.91823459529429089342384219791, −5.61833574147475309135447347180, −5.25395343888628502948518600511, −4.61077148196591260725709821754, −4.20844639577588816504211040783, −3.68660608995099879789749251284, −3.36261406321812250795690121311, −2.53043974442752146671270882425, −2.25443118451969417171892883918, −1.71168841250380650551263882796, −0.799135535478265123994517532385, 0, 0.799135535478265123994517532385, 1.71168841250380650551263882796, 2.25443118451969417171892883918, 2.53043974442752146671270882425, 3.36261406321812250795690121311, 3.68660608995099879789749251284, 4.20844639577588816504211040783, 4.61077148196591260725709821754, 5.25395343888628502948518600511, 5.61833574147475309135447347180, 5.91823459529429089342384219791, 6.34456479306805090859466531980, 6.49465019242782090984922319554, 7.14625534855191965904252856545

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