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Properties

Label 4-363e2-1.1-c1e2-0-7
Degree $4$
Conductor $131769$
Sign $1$
Analytic cond. $8.40170$
Root an. cond. $1.70251$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 8·5-s + 3·9-s − 16·15-s − 4·16-s + 4·23-s + 38·25-s − 4·27-s − 10·31-s + 6·37-s + 24·45-s + 4·47-s + 8·48-s − 13·49-s + 12·53-s − 20·59-s − 2·67-s − 8·69-s − 76·75-s − 32·80-s + 5·81-s + 24·89-s + 20·93-s + 10·97-s − 14·103-s − 12·111-s + 12·113-s + ⋯
L(s)  = 1  − 1.15·3-s + 3.57·5-s + 9-s − 4.13·15-s − 16-s + 0.834·23-s + 38/5·25-s − 0.769·27-s − 1.79·31-s + 0.986·37-s + 3.57·45-s + 0.583·47-s + 1.15·48-s − 1.85·49-s + 1.64·53-s − 2.60·59-s − 0.244·67-s − 0.963·69-s − 8.77·75-s − 3.57·80-s + 5/9·81-s + 2.54·89-s + 2.07·93-s + 1.01·97-s − 1.37·103-s − 1.13·111-s + 1.12·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(131769\)    =    \(3^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(8.40170\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 131769,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.153340679\)
\(L(\frac12)\) \(\approx\) \(2.153340679\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
5$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.5.ai_ba
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.19.a_bd
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.23.ae_by
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.31.k_dj
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.37.ag_df
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.a_acg
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.47.ae_du
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.59.u_ik
61$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.61.a_ej
67$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.67.c_ff
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.73.a_z
79$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.79.a_bl
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.89.ay_mk
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.97.ak_il
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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

−9.598150501746471856910168767253, −8.973480572648402934204682616447, −8.972019134073409502580254099142, −7.79298545455764963624452711488, −6.98676550080062361288693213459, −6.75395532990341575165448572479, −6.18537700457688272303978752564, −5.80045457911231193627474837008, −5.59539380126593624738687647200, −4.79085093103530693850477639954, −4.73167031165690126954598856077, −3.29412467945553736245288804676, −2.38116356225299620011633985249, −1.94003785572390378144466057337, −1.22560267668749840643223616905, 1.22560267668749840643223616905, 1.94003785572390378144466057337, 2.38116356225299620011633985249, 3.29412467945553736245288804676, 4.73167031165690126954598856077, 4.79085093103530693850477639954, 5.59539380126593624738687647200, 5.80045457911231193627474837008, 6.18537700457688272303978752564, 6.75395532990341575165448572479, 6.98676550080062361288693213459, 7.79298545455764963624452711488, 8.972019134073409502580254099142, 8.973480572648402934204682616447, 9.598150501746471856910168767253

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