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Properties

Label 4-81675-1.1-c1e2-0-8
Degree $4$
Conductor $81675$
Sign $-1$
Analytic cond. $5.20766$
Root an. cond. $1.51063$
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  + 2·2-s − 3-s − 4-s − 2·6-s − 8·8-s + 9-s + 4·11-s + 12-s − 7·16-s − 4·17-s + 2·18-s + 8·22-s + 8·24-s + 25-s − 27-s + 4·29-s + 14·32-s − 4·33-s − 8·34-s − 36-s − 20·37-s − 20·41-s − 4·44-s + 7·48-s − 14·49-s + 2·50-s + 4·51-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s − 1/2·4-s − 0.816·6-s − 2.82·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 7/4·16-s − 0.970·17-s + 0.471·18-s + 1.70·22-s + 1.63·24-s + 1/5·25-s − 0.192·27-s + 0.742·29-s + 2.47·32-s − 0.696·33-s − 1.37·34-s − 1/6·36-s − 3.28·37-s − 3.12·41-s − 0.603·44-s + 1.01·48-s − 2·49-s + 0.282·50-s + 0.560·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81675\)    =    \(3^{3} \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(5.20766\)
Root analytic conductor: \(1.51063\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 81675,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.2.ac_f
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
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

−9.724719918800934153926104184181, −8.803741276403576480309399510238, −8.554588868470699894187777077269, −8.307715053946613532072100459003, −6.90407371711147677445237484903, −6.76955885158912340897756950945, −6.32589394889552567488031990655, −5.52379452586844972509172293334, −5.04860090093668311608273372623, −4.82487869342099753310924019144, −4.03102032253133565798645985343, −3.63412818236731287070311550143, −3.05657152421448946850062568827, −1.66725869558357669408059166263, 0, 1.66725869558357669408059166263, 3.05657152421448946850062568827, 3.63412818236731287070311550143, 4.03102032253133565798645985343, 4.82487869342099753310924019144, 5.04860090093668311608273372623, 5.52379452586844972509172293334, 6.32589394889552567488031990655, 6.76955885158912340897756950945, 6.90407371711147677445237484903, 8.307715053946613532072100459003, 8.554588868470699894187777077269, 8.803741276403576480309399510238, 9.724719918800934153926104184181

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