Properties

Label 2-60984-1.1-c1-0-46
Degree $2$
Conductor $60984$
Sign $-1$
Analytic cond. $486.959$
Root an. cond. $22.0671$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s + 3·17-s − 6·19-s − 4·23-s − 4·25-s − 2·29-s + 10·31-s + 35-s − 2·37-s + 10·41-s + 3·43-s − 9·47-s + 49-s + 11·59-s − 5·67-s − 16·73-s + 16·79-s − 9·83-s + 3·85-s − 9·89-s − 6·95-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 0.727·17-s − 1.37·19-s − 0.834·23-s − 4/5·25-s − 0.371·29-s + 1.79·31-s + 0.169·35-s − 0.328·37-s + 1.56·41-s + 0.457·43-s − 1.31·47-s + 1/7·49-s + 1.43·59-s − 0.610·67-s − 1.87·73-s + 1.80·79-s − 0.987·83-s + 0.325·85-s − 0.953·89-s − 0.615·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60984\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(486.959\)
Root analytic conductor: \(22.0671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{60984} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 60984,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good5 \( 1 - T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.45188708396003, −14.20177830124236, −13.39088372356061, −13.22904422928038, −12.44321122775599, −12.09511285458270, −11.48341994246064, −11.05072618222168, −10.25579359276647, −10.11800728386530, −9.486361315008379, −8.870063719168695, −8.250320136132268, −7.939651165050863, −7.308059068376122, −6.563231344715518, −6.087204611093259, −5.670780422805176, −4.960510742610379, −4.271937421445530, −3.928157648658770, −2.986549513667232, −2.361516888610186, −1.779582508754257, −1.008632913036297, 0, 1.008632913036297, 1.779582508754257, 2.361516888610186, 2.986549513667232, 3.928157648658770, 4.271937421445530, 4.960510742610379, 5.670780422805176, 6.087204611093259, 6.563231344715518, 7.308059068376122, 7.939651165050863, 8.250320136132268, 8.870063719168695, 9.486361315008379, 10.11800728386530, 10.25579359276647, 11.05072618222168, 11.48341994246064, 12.09511285458270, 12.44321122775599, 13.22904422928038, 13.39088372356061, 14.20177830124236, 14.45188708396003

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