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Properties

Label 4-292032-1.1-c1e2-0-14
Degree $4$
Conductor $292032$
Sign $1$
Analytic cond. $18.6202$
Root an. cond. $2.07728$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 2·13-s + 16·19-s + 6·25-s − 27-s − 8·31-s + 12·37-s + 2·39-s + 8·43-s − 14·49-s − 16·57-s + 20·61-s − 8·67-s − 4·73-s − 6·75-s − 16·79-s + 81-s + 8·93-s − 20·97-s − 16·103-s + 20·109-s − 12·111-s − 2·117-s − 18·121-s + 127-s − 8·129-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.554·13-s + 3.67·19-s + 6/5·25-s − 0.192·27-s − 1.43·31-s + 1.97·37-s + 0.320·39-s + 1.21·43-s − 2·49-s − 2.11·57-s + 2.56·61-s − 0.977·67-s − 0.468·73-s − 0.692·75-s − 1.80·79-s + 1/9·81-s + 0.829·93-s − 2.03·97-s − 1.57·103-s + 1.91·109-s − 1.13·111-s − 0.184·117-s − 1.63·121-s + 0.0887·127-s − 0.704·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 292032 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 292032 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(292032\)    =    \(2^{6} \cdot 3^{3} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(18.6202\)
Root analytic conductor: \(2.07728\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 292032,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.579904327\)
\(L(\frac12)\) \(\approx\) \(1.579904327\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.19.aq_dy
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.a_ack
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.a_dy
59$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.59.a_ada
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.a_fi
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.83.a_abe
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−9.140403206898220946961227682758, −8.261470901569093424534734056213, −7.69837808741404231925046977179, −7.53138311391475412234846459345, −6.94953164128169925072485600653, −6.63559690348924214052369009469, −5.68484105112531909640928874037, −5.53392832912108757065282436695, −5.11677941303251687475460824152, −4.47584796632881532587650291705, −3.86619415183519071335286697657, −3.02936751855165763048811417152, −2.80021793098482472301348633140, −1.54183448393116236574503713957, −0.837203931274100821703489157823, 0.837203931274100821703489157823, 1.54183448393116236574503713957, 2.80021793098482472301348633140, 3.02936751855165763048811417152, 3.86619415183519071335286697657, 4.47584796632881532587650291705, 5.11677941303251687475460824152, 5.53392832912108757065282436695, 5.68484105112531909640928874037, 6.63559690348924214052369009469, 6.94953164128169925072485600653, 7.53138311391475412234846459345, 7.69837808741404231925046977179, 8.261470901569093424534734056213, 9.140403206898220946961227682758

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