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Properties

Label 4-156e2-1.1-c1e2-0-9
Degree $4$
Conductor $24336$
Sign $1$
Analytic cond. $1.55168$
Root an. cond. $1.11609$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 9-s − 2·13-s + 12·17-s − 6·23-s + 4·25-s − 4·27-s − 4·39-s − 8·43-s + 2·49-s + 24·51-s − 6·53-s − 4·61-s − 12·69-s + 8·75-s − 10·79-s − 11·81-s − 6·101-s − 20·103-s + 24·107-s + 12·113-s − 2·117-s − 14·121-s + 127-s − 16·129-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s − 0.554·13-s + 2.91·17-s − 1.25·23-s + 4/5·25-s − 0.769·27-s − 0.640·39-s − 1.21·43-s + 2/7·49-s + 3.36·51-s − 0.824·53-s − 0.512·61-s − 1.44·69-s + 0.923·75-s − 1.12·79-s − 1.22·81-s − 0.597·101-s − 1.97·103-s + 2.32·107-s + 1.12·113-s − 0.184·117-s − 1.27·121-s + 0.0887·127-s − 1.40·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(24336\)    =    \(2^{4} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1.55168\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 24336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.711774644\)
\(L(\frac12)\) \(\approx\) \(1.711774644\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.5.a_ae
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.g_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.a_k
41$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.41.a_i
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.47.a_c
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.g_bi
59$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.59.a_by
61$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.e_dm
67$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \) 2.67.a_dq
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.71.a_ck
73$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.73.a_aba
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.79.k_dy
83$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.83.a_acs
89$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.89.a_i
97$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.97.a_aeg
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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

−10.36539185412559528646758299291, −10.09361909012307865197488274015, −9.681794127214448782622884674313, −9.108449837706662154171637653416, −8.464821871272362020644116507815, −7.941297514258312253733864637354, −7.68729009799829749833130126622, −7.05205847457425997762496795297, −6.15498128319587771959340541202, −5.56268779024292319750686478621, −4.94385159638334824124403883982, −3.94105126425491521793657414786, −3.30728105064987180690084046019, −2.74276741871994897308823828763, −1.56981320253891731544646003058, 1.56981320253891731544646003058, 2.74276741871994897308823828763, 3.30728105064987180690084046019, 3.94105126425491521793657414786, 4.94385159638334824124403883982, 5.56268779024292319750686478621, 6.15498128319587771959340541202, 7.05205847457425997762496795297, 7.68729009799829749833130126622, 7.941297514258312253733864637354, 8.464821871272362020644116507815, 9.108449837706662154171637653416, 9.681794127214448782622884674313, 10.09361909012307865197488274015, 10.36539185412559528646758299291

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