Properties

Label 2-120e2-1.1-c1-0-89
Degree $2$
Conductor $14400$
Sign $-1$
Analytic cond. $114.984$
Root an. cond. $10.7230$
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  − 4·11-s − 2·13-s + 2·17-s − 4·19-s + 8·23-s + 6·29-s − 8·31-s + 6·37-s + 6·41-s − 4·43-s − 7·49-s + 2·53-s − 4·59-s + 2·61-s + 4·67-s + 8·71-s − 10·73-s + 8·79-s − 4·83-s + 6·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s + 1.11·29-s − 1.43·31-s + 0.986·37-s + 0.937·41-s − 0.609·43-s − 49-s + 0.274·53-s − 0.520·59-s + 0.256·61-s + 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.900·79-s − 0.439·83-s + 0.635·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(114.984\)
Root analytic conductor: \(10.7230\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{14400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 14400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.40723713140756, −15.86339438152485, −15.18368519819409, −14.77093452268043, −14.30234101084824, −13.47172081259227, −12.89260027578312, −12.68991299830959, −11.95036362627315, −11.05203856476603, −10.87622806389476, −10.12434474671448, −9.578990932597495, −8.903055293706380, −8.255796980767851, −7.677362533380482, −7.112325659768094, −6.423435442961949, −5.649879426782975, −5.010032767020927, −4.548653159077549, −3.537100377471948, −2.805684776949022, −2.218456936948766, −1.081774835208433, 0, 1.081774835208433, 2.218456936948766, 2.805684776949022, 3.537100377471948, 4.548653159077549, 5.010032767020927, 5.649879426782975, 6.423435442961949, 7.112325659768094, 7.677362533380482, 8.255796980767851, 8.903055293706380, 9.578990932597495, 10.12434474671448, 10.87622806389476, 11.05203856476603, 11.95036362627315, 12.68991299830959, 12.89260027578312, 13.47172081259227, 14.30234101084824, 14.77093452268043, 15.18368519819409, 15.86339438152485, 16.40723713140756

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