Properties

Label 2-64400-1.1-c1-0-28
Degree $2$
Conductor $64400$
Sign $-1$
Analytic cond. $514.236$
Root an. cond. $22.6767$
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  + 7-s − 3·9-s − 4·11-s − 6·13-s + 2·17-s − 4·19-s − 23-s − 2·29-s + 4·31-s + 2·37-s − 6·41-s + 12·43-s − 12·47-s + 49-s + 10·53-s + 2·61-s − 3·63-s + 12·67-s − 8·71-s + 14·73-s − 4·77-s − 8·79-s + 9·81-s − 4·83-s + 6·89-s − 6·91-s + 10·97-s + ⋯
L(s)  = 1  + 0.377·7-s − 9-s − 1.20·11-s − 1.66·13-s + 0.485·17-s − 0.917·19-s − 0.208·23-s − 0.371·29-s + 0.718·31-s + 0.328·37-s − 0.937·41-s + 1.82·43-s − 1.75·47-s + 1/7·49-s + 1.37·53-s + 0.256·61-s − 0.377·63-s + 1.46·67-s − 0.949·71-s + 1.63·73-s − 0.455·77-s − 0.900·79-s + 81-s − 0.439·83-s + 0.635·89-s − 0.628·91-s + 1.01·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64400\)    =    \(2^{4} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(514.236\)
Root analytic conductor: \(22.6767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{64400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 64400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.50389226871077, −14.15188921666803, −13.41176504022586, −13.01032132490409, −12.39544369683273, −12.02909760670706, −11.46735967137142, −10.93513433244147, −10.44730847848391, −9.874835293237082, −9.549488106865449, −8.625142984552253, −8.375742142741619, −7.722034781453654, −7.412656091223387, −6.659795091448737, −6.010439653738106, −5.406069828135540, −5.022100534290571, −4.498660119147399, −3.683611508519223, −2.892740435684032, −2.429859116377826, −1.974887118450823, −0.7112500429186889, 0, 0.7112500429186889, 1.974887118450823, 2.429859116377826, 2.892740435684032, 3.683611508519223, 4.498660119147399, 5.022100534290571, 5.406069828135540, 6.010439653738106, 6.659795091448737, 7.412656091223387, 7.722034781453654, 8.375742142741619, 8.625142984552253, 9.549488106865449, 9.874835293237082, 10.44730847848391, 10.93513433244147, 11.46735967137142, 12.02909760670706, 12.39544369683273, 13.01032132490409, 13.41176504022586, 14.15188921666803, 14.50389226871077

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