Properties

Label 4-1710e2-1.1-c1e2-0-8
Degree $4$
Conductor $2924100$
Sign $1$
Analytic cond. $186.443$
Root an. cond. $3.69518$
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 − 5-s − 2·7-s + 8-s + 10-s − 12·11-s − 5·13-s + 2·14-s − 16-s + 8·19-s + 12·22-s + 6·23-s + 5·26-s + 6·29-s + 10·31-s + 2·35-s + 22·37-s − 8·38-s − 40-s + 6·41-s + 43-s − 6·46-s + 12·47-s − 11·49-s − 12·53-s + 12·55-s − 2·56-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.447·5-s − 0.755·7-s + 0.353·8-s + 0.316·10-s − 3.61·11-s − 1.38·13-s + 0.534·14-s − 1/4·16-s + 1.83·19-s + 2.55·22-s + 1.25·23-s + 0.980·26-s + 1.11·29-s + 1.79·31-s + 0.338·35-s + 3.61·37-s − 1.29·38-s − 0.158·40-s + 0.937·41-s + 0.152·43-s − 0.884·46-s + 1.75·47-s − 1.57·49-s − 1.64·53-s + 1.61·55-s − 0.267·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2924100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(186.443\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1710} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2924100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9303225768\)
\(L(\frac12)\) \(\approx\) \(0.9303225768\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−9.531831806814563650080064999592, −9.523171109724453074766595188422, −8.424494293813770971897252856811, −8.406408255314473290418651217003, −7.88089015461384895756798211898, −7.59449629469905035028858737949, −7.40450331673685842392819858653, −7.07509273453697507673996152864, −6.21191097719726046694913104037, −5.99418914435065781625643780136, −5.35310012251550029561371903598, −4.88150868462089107770567443747, −4.81600566699623103743042237138, −4.33184466067698072620885736122, −3.23123302787833580589793765112, −2.95398500532572820501256108502, −2.61796221773856478868216534150, −2.32481392672750532565795428611, −0.71703141297952087308926020841, −0.67110736924565267610648496899, 0.67110736924565267610648496899, 0.71703141297952087308926020841, 2.32481392672750532565795428611, 2.61796221773856478868216534150, 2.95398500532572820501256108502, 3.23123302787833580589793765112, 4.33184466067698072620885736122, 4.81600566699623103743042237138, 4.88150868462089107770567443747, 5.35310012251550029561371903598, 5.99418914435065781625643780136, 6.21191097719726046694913104037, 7.07509273453697507673996152864, 7.40450331673685842392819858653, 7.59449629469905035028858737949, 7.88089015461384895756798211898, 8.406408255314473290418651217003, 8.424494293813770971897252856811, 9.523171109724453074766595188422, 9.531831806814563650080064999592

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