Properties

Label 2-94640-1.1-c1-0-34
Degree $2$
Conductor $94640$
Sign $-1$
Analytic cond. $755.704$
Root an. cond. $27.4900$
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  − 3·3-s + 5-s − 7-s + 6·9-s − 5·11-s − 3·15-s − 17-s + 6·19-s + 3·21-s − 6·23-s + 25-s − 9·27-s − 9·29-s − 4·31-s + 15·33-s − 35-s − 2·37-s + 4·41-s − 10·43-s + 6·45-s − 47-s + 49-s + 3·51-s + 4·53-s − 5·55-s − 18·57-s − 8·59-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.447·5-s − 0.377·7-s + 2·9-s − 1.50·11-s − 0.774·15-s − 0.242·17-s + 1.37·19-s + 0.654·21-s − 1.25·23-s + 1/5·25-s − 1.73·27-s − 1.67·29-s − 0.718·31-s + 2.61·33-s − 0.169·35-s − 0.328·37-s + 0.624·41-s − 1.52·43-s + 0.894·45-s − 0.145·47-s + 1/7·49-s + 0.420·51-s + 0.549·53-s − 0.674·55-s − 2.38·57-s − 1.04·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 94640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(94640\)    =    \(2^{4} \cdot 5 \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(755.704\)
Root analytic conductor: \(27.4900\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{94640} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 94640,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 \)
good3 \( 1 + p T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−13.87295423902826, −13.37739144003778, −13.02243284143478, −12.52968059176288, −12.11928026171005, −11.51680674429592, −11.12242837389310, −10.68839658209853, −10.17456804862625, −9.728810935815525, −9.439442213780493, −8.513359163995569, −7.873827204529919, −7.271987717819208, −7.048969935921668, −6.158143371933241, −5.849032191104675, −5.398012254926677, −5.046248531758572, −4.400129984306933, −3.633680509562861, −3.020931523271220, −2.084391402739923, −1.596741104142677, −0.5805540485400794, 0, 0.5805540485400794, 1.596741104142677, 2.084391402739923, 3.020931523271220, 3.633680509562861, 4.400129984306933, 5.046248531758572, 5.398012254926677, 5.849032191104675, 6.158143371933241, 7.048969935921668, 7.271987717819208, 7.873827204529919, 8.513359163995569, 9.439442213780493, 9.728810935815525, 10.17456804862625, 10.68839658209853, 11.12242837389310, 11.51680674429592, 12.11928026171005, 12.52968059176288, 13.02243284143478, 13.37739144003778, 13.87295423902826

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