Properties

Label 4-67712-1.1-c1e2-0-2
Degree $4$
Conductor $67712$
Sign $1$
Analytic cond. $4.31737$
Root an. cond. $1.44146$
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-s + 4-s − 8-s − 6·9-s + 4·11-s + 16-s − 4·17-s + 6·18-s − 4·19-s − 4·22-s + 6·25-s − 32-s + 4·34-s − 6·36-s + 4·38-s + 12·41-s + 20·43-s + 4·44-s + 2·49-s − 6·50-s + 24·59-s + 64-s − 20·67-s − 4·68-s + 6·72-s + 12·73-s − 4·76-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2·9-s + 1.20·11-s + 1/4·16-s − 0.970·17-s + 1.41·18-s − 0.917·19-s − 0.852·22-s + 6/5·25-s − 0.176·32-s + 0.685·34-s − 36-s + 0.648·38-s + 1.87·41-s + 3.04·43-s + 0.603·44-s + 2/7·49-s − 0.848·50-s + 3.12·59-s + 1/8·64-s − 2.44·67-s − 0.485·68-s + 0.707·72-s + 1.40·73-s − 0.458·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(67712\)    =    \(2^{7} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(4.31737\)
Root analytic conductor: \(1.44146\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 67712,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8497979687\)
\(L(\frac12)\) \(\approx\) \(0.8497979687\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( 1 + T \)
23$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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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

−9.594811427439617650906242640583, −9.130106351311024965413041321010, −8.971935452602447588346290189052, −8.530890794481804115364421751679, −7.987382994028197640535077252881, −7.36808262548174147697981397779, −6.69441913641732873591204403077, −6.27306507536673917111849783164, −5.84432417959727908710252168629, −5.18657588517648102575315063393, −4.26966867388668012719554112067, −3.74569573400474523501557037674, −2.63232737474552247848840602025, −2.38824907079698926735327113113, −0.819637353542681802783519296171, 0.819637353542681802783519296171, 2.38824907079698926735327113113, 2.63232737474552247848840602025, 3.74569573400474523501557037674, 4.26966867388668012719554112067, 5.18657588517648102575315063393, 5.84432417959727908710252168629, 6.27306507536673917111849783164, 6.69441913641732873591204403077, 7.36808262548174147697981397779, 7.987382994028197640535077252881, 8.530890794481804115364421751679, 8.971935452602447588346290189052, 9.130106351311024965413041321010, 9.594811427439617650906242640583

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