↑

Properties

Label 4-363e2-1.1-c1e2-0-4
Degree $4$
Conductor $131769$
Sign $-1$
Analytic cond. $8.40170$
Root an. cond. $1.70251$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·4-s + 9-s + 6·12-s + 5·16-s − 9·25-s + 4·27-s + 4·31-s − 3·36-s + 6·37-s − 10·48-s + 10·49-s − 3·64-s − 4·67-s + 18·75-s − 11·81-s − 8·93-s − 26·97-s + 27·100-s + 16·103-s − 12·108-s − 12·111-s − 12·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1.15·3-s − 3/2·4-s + 1/3·9-s + 1.73·12-s + 5/4·16-s − 9/5·25-s + 0.769·27-s + 0.718·31-s − 1/2·36-s + 0.986·37-s − 1.44·48-s + 10/7·49-s − 3/8·64-s − 0.488·67-s + 2.07·75-s − 1.22·81-s − 0.829·93-s − 2.63·97-s + 2.69·100-s + 1.57·103-s − 1.15·108-s − 1.13·111-s − 1.07·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 131769,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad3$C_2$ \( 1 + 2 T + p T^{2} \)
11 \( 1 \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.2.a_d
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.a_j
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.13.a_az
17$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.17.a_j
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.23.a_bq
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.a_ax
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.37.ag_df
41$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.41.a_cf
43$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.43.a_adi
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.a_dm
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.a_z
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.61.a_adi
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.67.e_fi
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.73.a_afm
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.79.a_acg
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.a_dt
97$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.97.ba_nz
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.281143958638857692960467551261, −8.634054297830232023940201708085, −8.249977975353377798009088871094, −7.75592585898703081972787164653, −7.15112546908013756778030772487, −6.48577630694408373513070381672, −5.86606203787301044654210886361, −5.64630551311784916090773833238, −5.04164591351084708065846963538, −4.41303363846825067569218781441, −4.14650031509247077700092409613, −3.35443968705909100459094716436, −2.36213852699006213104899542481, −1.05049044658149268370353805033, 0, 1.05049044658149268370353805033, 2.36213852699006213104899542481, 3.35443968705909100459094716436, 4.14650031509247077700092409613, 4.41303363846825067569218781441, 5.04164591351084708065846963538, 5.64630551311784916090773833238, 5.86606203787301044654210886361, 6.48577630694408373513070381672, 7.15112546908013756778030772487, 7.75592585898703081972787164653, 8.249977975353377798009088871094, 8.634054297830232023940201708085, 9.281143958638857692960467551261

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