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Properties

Label 4-486e2-1.1-c1e2-0-5
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 $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 8·7-s + 4·13-s + 16-s − 2·19-s − 25-s − 8·28-s + 4·31-s + 16·37-s − 8·43-s + 34·49-s + 4·52-s + 16·61-s + 64-s − 26·67-s − 14·73-s − 2·76-s − 8·79-s − 32·91-s − 26·97-s − 100-s − 8·103-s − 20·109-s − 8·112-s + 14·121-s + 4·124-s + 127-s + ⋯
L(s)  = 1  + 1/2·4-s − 3.02·7-s + 1.10·13-s + 1/4·16-s − 0.458·19-s − 1/5·25-s − 1.51·28-s + 0.718·31-s + 2.63·37-s − 1.21·43-s + 34/7·49-s + 0.554·52-s + 2.04·61-s + 1/8·64-s − 3.17·67-s − 1.63·73-s − 0.229·76-s − 0.900·79-s − 3.35·91-s − 2.63·97-s − 0.0999·100-s − 0.788·103-s − 1.91·109-s − 0.755·112-s + 1.27·121-s + 0.359·124-s + 0.0887·127-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: \(1\)
Selberg data: \((4,\ 236196,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.29.a_bx
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.37.aq_fi
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.i_dy
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.47.a_dh
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.a_z
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.67.ba_lr
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.71.a_cj
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.73.o_hn
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.97.ba_nz
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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

−8.679568130561498556922581017813, −8.461723979708796657095583439123, −7.73731656802671746427138225950, −7.08955181338173010150793512782, −6.77100901767751662201104029983, −6.32716563151636225311285217480, −5.87454672168208509315529636024, −5.82116351609837137927962809945, −4.60568349400255175327689787418, −3.96642516208177921278290901157, −3.54926944648476867148369691792, −2.77728629591243829342757204382, −2.71135029910568398285484502843, −1.26276179722608830528402176097, 0, 1.26276179722608830528402176097, 2.71135029910568398285484502843, 2.77728629591243829342757204382, 3.54926944648476867148369691792, 3.96642516208177921278290901157, 4.60568349400255175327689787418, 5.82116351609837137927962809945, 5.87454672168208509315529636024, 6.32716563151636225311285217480, 6.77100901767751662201104029983, 7.08955181338173010150793512782, 7.73731656802671746427138225950, 8.461723979708796657095583439123, 8.679568130561498556922581017813

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