Properties

Label 4-586971-1.1-c1e2-0-5
Degree $4$
Conductor $586971$
Sign $1$
Analytic cond. $37.4257$
Root an. cond. $2.47339$
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  + 3-s − 4·4-s − 2·9-s + 11-s − 4·12-s + 12·16-s + 12·17-s − 25-s − 5·27-s + 12·29-s + 10·31-s + 33-s + 8·36-s + 22·37-s − 12·41-s − 4·44-s + 12·48-s + 49-s + 12·51-s − 32·64-s + 10·67-s − 48·68-s − 75-s + 81-s − 24·83-s + 12·87-s + 10·93-s + ⋯
L(s)  = 1  + 0.577·3-s − 2·4-s − 2/3·9-s + 0.301·11-s − 1.15·12-s + 3·16-s + 2.91·17-s − 1/5·25-s − 0.962·27-s + 2.22·29-s + 1.79·31-s + 0.174·33-s + 4/3·36-s + 3.61·37-s − 1.87·41-s − 0.603·44-s + 1.73·48-s + 1/7·49-s + 1.68·51-s − 4·64-s + 1.22·67-s − 5.82·68-s − 0.115·75-s + 1/9·81-s − 2.63·83-s + 1.28·87-s + 1.03·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(586971\)    =    \(3^{2} \cdot 7^{2} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(37.4257\)
Root analytic conductor: \(2.47339\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 586971,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.679171307\)
\(L(\frac12)\) \(\approx\) \(1.679171307\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 - T + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−8.508833833811404945218630536704, −8.085327774959512351361821751845, −7.901671452709333870689471023015, −7.32024742481712425147757589138, −6.46084267575588440942544784885, −5.81865396854508245636334597208, −5.80492978659389909733109582978, −4.95929836259470425310107953583, −4.73710013360513636713787424546, −4.10975481209818427421549838760, −3.61592205748787980286928494613, −3.01251424255196705097668786615, −2.75685822824512438972431635156, −1.19698223664384976616389110079, −0.821355075233086309329735836862, 0.821355075233086309329735836862, 1.19698223664384976616389110079, 2.75685822824512438972431635156, 3.01251424255196705097668786615, 3.61592205748787980286928494613, 4.10975481209818427421549838760, 4.73710013360513636713787424546, 4.95929836259470425310107953583, 5.80492978659389909733109582978, 5.81865396854508245636334597208, 6.46084267575588440942544784885, 7.32024742481712425147757589138, 7.901671452709333870689471023015, 8.085327774959512351361821751845, 8.508833833811404945218630536704

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