↑

Properties

Label 4-363e2-1.1-c1e2-0-14
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 $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s − 6·5-s + 3·9-s + 2·12-s + 12·15-s − 3·16-s + 6·20-s − 12·23-s + 17·25-s − 4·27-s + 8·31-s − 3·36-s − 22·37-s − 18·45-s + 6·48-s − 2·49-s − 18·53-s − 12·59-s − 12·60-s + 7·64-s − 4·67-s + 24·69-s − 12·71-s − 34·75-s + 18·80-s + 5·81-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s − 2.68·5-s + 9-s + 0.577·12-s + 3.09·15-s − 3/4·16-s + 1.34·20-s − 2.50·23-s + 17/5·25-s − 0.769·27-s + 1.43·31-s − 1/2·36-s − 3.61·37-s − 2.68·45-s + 0.866·48-s − 2/7·49-s − 2.47·53-s − 1.56·59-s − 1.54·60-s + 7/8·64-s − 0.488·67-s + 2.88·69-s − 1.42·71-s − 3.92·75-s + 2.01·80-s + 5/9·81-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: \(2\)
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$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \) 2.13.a_x
17$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.17.a_bf
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.29.a_cd
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.37.w_hn
41$C_2^2$ \( 1 + 79 T^{2} + p^{2} T^{4} \) 2.41.a_db
43$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.43.a_cw
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.59.m_fy
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.67.e_fi
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.73.a_du
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.89.as_jz
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.97.o_jj
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

−11.48613840251284076179398338363, −10.88929931871862449135005452623, −10.28328728028678558856477572353, −10.17555670712698730983302805703, −9.279137318937476680564254609479, −8.789592707949681467230495188896, −8.233106999068815384359466088835, −7.83888142451841888077368799506, −7.59634104912019193486672986525, −6.90239902824418094849867412203, −6.45942841117226412127905637665, −5.92665930083780474844521723709, −4.92362723553219248906923643921, −4.78736398400057893106922846764, −4.08874146042552895871178962658, −3.81710189102304683427755985034, −3.14712099992739018433219472729, −1.70975558990753806142579652840, 0, 0, 1.70975558990753806142579652840, 3.14712099992739018433219472729, 3.81710189102304683427755985034, 4.08874146042552895871178962658, 4.78736398400057893106922846764, 4.92362723553219248906923643921, 5.92665930083780474844521723709, 6.45942841117226412127905637665, 6.90239902824418094849867412203, 7.59634104912019193486672986525, 7.83888142451841888077368799506, 8.233106999068815384359466088835, 8.789592707949681467230495188896, 9.279137318937476680564254609479, 10.17555670712698730983302805703, 10.28328728028678558856477572353, 10.88929931871862449135005452623, 11.48613840251284076179398338363

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