Properties

Label 4-65219-1.1-c1e2-0-0
Degree $4$
Conductor $65219$
Sign $1$
Analytic cond. $4.15841$
Root an. cond. $1.42801$
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 − 4·4-s + 6·5-s − 3·9-s − 11-s − 8·12-s + 12·15-s + 12·16-s − 24·20-s + 6·23-s + 17·25-s − 14·27-s + 10·31-s − 2·33-s + 12·36-s + 22·37-s + 4·44-s − 18·45-s + 24·48-s + 49-s − 12·53-s − 6·55-s − 18·59-s − 48·60-s − 32·64-s + 10·67-s + 12·69-s + ⋯
L(s)  = 1  + 1.15·3-s − 2·4-s + 2.68·5-s − 9-s − 0.301·11-s − 2.30·12-s + 3.09·15-s + 3·16-s − 5.36·20-s + 1.25·23-s + 17/5·25-s − 2.69·27-s + 1.79·31-s − 0.348·33-s + 2·36-s + 3.61·37-s + 0.603·44-s − 2.68·45-s + 3.46·48-s + 1/7·49-s − 1.64·53-s − 0.809·55-s − 2.34·59-s − 6.19·60-s − 4·64-s + 1.22·67-s + 1.44·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(65219\)    =    \(7^{2} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(4.15841\)
Root analytic conductor: \(1.42801\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 65219,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.933947068\)
\(L(\frac12)\) \(\approx\) \(1.933947068\)
\(L(\frac{3}{2})\) 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)$
bad7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
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

−9.594904136117515823873522897188, −9.487459835858535121288790482109, −9.121939609810073781754575230776, −8.508833833811404945218630536704, −8.085327774959512351361821751845, −7.79543673470654027074280114997, −6.46084267575588440942544784885, −5.89237731397243587720148634695, −5.81865396854508245636334597208, −4.95929836259470425310107953583, −4.69940224529243356978622440045, −3.61592205748787980286928494613, −2.75685822824512438972431635156, −2.52489627324288051568453078479, −1.19698223664384976616389110079, 1.19698223664384976616389110079, 2.52489627324288051568453078479, 2.75685822824512438972431635156, 3.61592205748787980286928494613, 4.69940224529243356978622440045, 4.95929836259470425310107953583, 5.81865396854508245636334597208, 5.89237731397243587720148634695, 6.46084267575588440942544784885, 7.79543673470654027074280114997, 8.085327774959512351361821751845, 8.508833833811404945218630536704, 9.121939609810073781754575230776, 9.487459835858535121288790482109, 9.594904136117515823873522897188

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