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Properties

Label 4-924e2-1.1-c1e2-0-49
Degree $4$
Conductor $853776$
Sign $1$
Analytic cond. $54.4374$
Root an. cond. $2.71628$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·7-s − 2·9-s − 8·13-s − 4·19-s + 2·21-s − 9·25-s + 5·27-s − 2·31-s − 18·37-s + 8·39-s + 16·43-s + 3·49-s + 4·57-s − 4·61-s + 4·63-s + 22·67-s − 28·73-s + 9·75-s − 28·79-s + 81-s + 16·91-s + 2·93-s − 18·97-s − 8·103-s + 8·109-s + 18·111-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s − 2/3·9-s − 2.21·13-s − 0.917·19-s + 0.436·21-s − 9/5·25-s + 0.962·27-s − 0.359·31-s − 2.95·37-s + 1.28·39-s + 2.43·43-s + 3/7·49-s + 0.529·57-s − 0.512·61-s + 0.503·63-s + 2.68·67-s − 3.27·73-s + 1.03·75-s − 3.15·79-s + 1/9·81-s + 1.67·91-s + 0.207·93-s − 1.82·97-s − 0.788·103-s + 0.766·109-s + 1.70·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 853776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 853776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(853776\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(54.4374\)
Root analytic conductor: \(2.71628\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 853776,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_2$ \( 1 + T + p T^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.a_j
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.23.a_bt
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.31.c_cl
37$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.37.s_fz
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
59$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.59.a_en
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.67.aw_jv
71$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.71.a_v
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.73.bc_ne
79$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.79.bc_nq
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.89.a_j
97$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.97.s_kp
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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

−7.55051318247375673767362801689, −7.20044051165186333263902429412, −7.07506627295383549778258913254, −6.33749430761827790522533904144, −5.93373339407611122896315603521, −5.50454839200369076494692983370, −5.18672240750960531371029490123, −4.53314310994402234388074278431, −4.05589871533530008614715770615, −3.51127816042162066284861488111, −2.62443093669091058806415890195, −2.50570574661996799886651591918, −1.61001524210966283725621308037, 0, 0, 1.61001524210966283725621308037, 2.50570574661996799886651591918, 2.62443093669091058806415890195, 3.51127816042162066284861488111, 4.05589871533530008614715770615, 4.53314310994402234388074278431, 5.18672240750960531371029490123, 5.50454839200369076494692983370, 5.93373339407611122896315603521, 6.33749430761827790522533904144, 7.07506627295383549778258913254, 7.20044051165186333263902429412, 7.55051318247375673767362801689

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