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Properties

Label 4-486e2-1.1-c1e2-0-3
Degree $4$
Conductor $236196$
Sign $1$
Analytic cond. $15.0600$
Root an. cond. $1.96995$
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  + 2-s + 3·5-s − 2·7-s − 8-s + 3·10-s + 4·13-s − 2·14-s − 16-s + 12·17-s − 14·19-s + 9·23-s + 5·25-s + 4·26-s + 9·29-s − 2·31-s + 12·34-s − 6·35-s − 8·37-s − 14·38-s − 3·40-s − 6·41-s + 4·43-s + 9·46-s + 3·47-s + 7·49-s + 5·50-s + 6·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.34·5-s − 0.755·7-s − 0.353·8-s + 0.948·10-s + 1.10·13-s − 0.534·14-s − 1/4·16-s + 2.91·17-s − 3.21·19-s + 1.87·23-s + 25-s + 0.784·26-s + 1.67·29-s − 0.359·31-s + 2.05·34-s − 1.01·35-s − 1.31·37-s − 2.27·38-s − 0.474·40-s − 0.937·41-s + 0.609·43-s + 1.32·46-s + 0.437·47-s + 49-s + 0.707·50-s + 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 236196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236196 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(236196\)    =    \(2^{2} \cdot 3^{10}\)
Sign: $1$
Analytic conductor: \(15.0600\)
Root analytic conductor: \(1.96995\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 236196,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.083067731\)
\(L(\frac12)\) \(\approx\) \(3.083067731\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_e
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ad
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_d
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.19.o_dj
23$C_2^2$ \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.23.aj_cg
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_ca
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_abb
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_af
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_abb
47$C_2^2$ \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.47.ad_abm
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.53.ag_el
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.61.ak_bn
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.f_abq
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.71.g_fv
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.73.o_hn
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ap
83$C_2^2$ \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_abv
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.89.y_mk
97$C_2^2$ \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) 2.97.ab_ads
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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

−11.03457261891200222356840903600, −10.64557171912236157911321979084, −10.20026464587774252493112762676, −10.10754235561458609037512342213, −9.530771136615603992571230523222, −8.766045867232726927570207969390, −8.563981486422708538670718926232, −8.438662557282255484760827983418, −7.14210723258201388684837341199, −7.12040862029975227921916369845, −6.21896888550664762617002940327, −6.17423520285278835618122467941, −5.52511068690168711259613773189, −5.28175130366550267513058917458, −4.40894079072760182883170848310, −3.96350899921876026263251702378, −3.09445757754292285123753254422, −2.93010306679192687358797931440, −1.89102821128581011153876899378, −1.05188513745075441533185082570, 1.05188513745075441533185082570, 1.89102821128581011153876899378, 2.93010306679192687358797931440, 3.09445757754292285123753254422, 3.96350899921876026263251702378, 4.40894079072760182883170848310, 5.28175130366550267513058917458, 5.52511068690168711259613773189, 6.17423520285278835618122467941, 6.21896888550664762617002940327, 7.12040862029975227921916369845, 7.14210723258201388684837341199, 8.438662557282255484760827983418, 8.563981486422708538670718926232, 8.766045867232726927570207969390, 9.530771136615603992571230523222, 10.10754235561458609037512342213, 10.20026464587774252493112762676, 10.64557171912236157911321979084, 11.03457261891200222356840903600

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