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Properties

Label 4-363e2-1.1-c1e2-0-11
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4-s + 4·5-s + 3·9-s + 2·12-s + 8·15-s − 3·16-s + 4·20-s − 8·23-s + 2·25-s + 4·27-s + 3·36-s + 4·37-s + 12·45-s + 16·47-s − 6·48-s + 6·49-s + 12·53-s + 8·60-s − 7·64-s − 24·67-s − 16·69-s − 16·71-s + 4·75-s − 12·80-s + 5·81-s − 28·89-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s + 1.78·5-s + 9-s + 0.577·12-s + 2.06·15-s − 3/4·16-s + 0.894·20-s − 1.66·23-s + 2/5·25-s + 0.769·27-s + 1/2·36-s + 0.657·37-s + 1.78·45-s + 2.33·47-s − 0.866·48-s + 6/7·49-s + 1.64·53-s + 1.03·60-s − 7/8·64-s − 2.93·67-s − 1.92·69-s − 1.89·71-s + 0.461·75-s − 1.34·80-s + 5/9·81-s − 2.96·89-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

\(L(1)\) \(\approx\) \(3.605239194\)
\(L(\frac12)\) \(\approx\) \(3.605239194\)
\(L(\frac{3}{2})\) 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_ab
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.5.ae_o
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.19.a_s
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.29.a_bm
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.41.a_ck
43$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.43.a_co
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.61.a_bq
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.73.a_co
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.79.a_aw
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.83.a_di
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.89.bc_ok
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−11.96813214338545798912435604191, −11.09872188734298677461845321186, −10.38104632610696553786721885962, −10.29874149462326644647336749676, −9.800274452758984125650845855443, −9.403921354606738867216501418016, −8.768845110335529202437101245554, −8.753138004800805002071256031689, −7.899713085368000972535923301565, −7.23339165975832432669132254947, −7.21034934891176342175331080954, −6.21714992398416050933747042136, −5.78864244709404609915381922195, −5.73387068866665816679127942391, −4.44686979466227102752556716323, −4.23324492548299928706192072988, −3.27526656858308043035087270177, −2.34514472825920689329196705733, −2.30867629089593721766915328887, −1.50398825572931132120664116879, 1.50398825572931132120664116879, 2.30867629089593721766915328887, 2.34514472825920689329196705733, 3.27526656858308043035087270177, 4.23324492548299928706192072988, 4.44686979466227102752556716323, 5.73387068866665816679127942391, 5.78864244709404609915381922195, 6.21714992398416050933747042136, 7.21034934891176342175331080954, 7.23339165975832432669132254947, 7.899713085368000972535923301565, 8.753138004800805002071256031689, 8.768845110335529202437101245554, 9.403921354606738867216501418016, 9.800274452758984125650845855443, 10.29874149462326644647336749676, 10.38104632610696553786721885962, 11.09872188734298677461845321186, 11.96813214338545798912435604191

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