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Properties

Label 4-57e3-1.1-c1e2-0-3
Degree $4$
Conductor $185193$
Sign $1$
Analytic cond. $11.8080$
Root an. cond. $1.85372$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 3-s + 8·4-s + 4·6-s − 10·7-s − 8·8-s + 9-s − 8·12-s + 40·14-s − 4·16-s − 4·18-s − 19-s + 10·21-s + 8·24-s − 25-s − 27-s − 80·28-s − 4·29-s + 32·32-s + 8·36-s + 4·38-s − 40·42-s − 2·43-s + 4·48-s + 61·49-s + 4·50-s + 20·53-s + ⋯
L(s)  = 1  − 2.82·2-s − 0.577·3-s + 4·4-s + 1.63·6-s − 3.77·7-s − 2.82·8-s + 1/3·9-s − 2.30·12-s + 10.6·14-s − 16-s − 0.942·18-s − 0.229·19-s + 2.18·21-s + 1.63·24-s − 1/5·25-s − 0.192·27-s − 15.1·28-s − 0.742·29-s + 5.65·32-s + 4/3·36-s + 0.648·38-s − 6.17·42-s − 0.304·43-s + 0.577·48-s + 61/7·49-s + 0.565·50-s + 2.74·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(185193\)    =    \(3^{3} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(11.8080\)
Root analytic conductor: \(1.85372\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 185193,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad3$C_1$ \( 1 + T \)
19$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 + p T + p T^{2} )^{2} \) 2.2.e_i
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.7.k_bn
11$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.11.a_v
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.17.a_bh
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.a_ba
37$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.37.a_cw
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.53.au_hy
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.59.q_ha
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.61.c_et
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.a_cs
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.73.w_kh
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.a_dq
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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

−8.836427381447004052087018681826, −8.680448170254320431881606216967, −7.47848778675296589128891353641, −7.42573159443448293402645158360, −7.00357033526907285371973300329, −6.41173993136661837405396202601, −6.15488117667979991166374049282, −5.61647921542357100461421630772, −4.40039433035204962000756580058, −3.81591641511052084615077615666, −3.00491682168355671002496814154, −2.44349751141625690337537557236, −1.22456142142336429144535327783, 0, 0, 1.22456142142336429144535327783, 2.44349751141625690337537557236, 3.00491682168355671002496814154, 3.81591641511052084615077615666, 4.40039433035204962000756580058, 5.61647921542357100461421630772, 6.15488117667979991166374049282, 6.41173993136661837405396202601, 7.00357033526907285371973300329, 7.42573159443448293402645158360, 7.47848778675296589128891353641, 8.680448170254320431881606216967, 8.836427381447004052087018681826

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