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Properties

Label 2-15680-1.1-c1-0-11
Degree $2$
Conductor $15680$
Sign $1$
Analytic cond. $125.205$
Root an. cond. $11.1895$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 2·9-s − 3·11-s − 13-s − 15-s − 5·17-s + 6·19-s + 25-s − 5·27-s + 5·29-s + 2·31-s − 3·33-s + 4·37-s − 39-s − 2·41-s − 10·43-s + 2·45-s − 9·47-s − 5·51-s − 6·53-s + 3·55-s + 6·57-s + 6·59-s + 12·61-s + 65-s + 2·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.904·11-s − 0.277·13-s − 0.258·15-s − 1.21·17-s + 1.37·19-s + 1/5·25-s − 0.962·27-s + 0.928·29-s + 0.359·31-s − 0.522·33-s + 0.657·37-s − 0.160·39-s − 0.312·41-s − 1.52·43-s + 0.298·45-s − 1.31·47-s − 0.700·51-s − 0.824·53-s + 0.404·55-s + 0.794·57-s + 0.781·59-s + 1.53·61-s + 0.124·65-s + 0.244·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15680\)    =    \(2^{6} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(125.205\)
Root analytic conductor: \(11.1895\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.363121147\)
\(L(\frac12)\) \(\approx\) \(1.363121147\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−16.00666505856340, −15.40391328411178, −14.92591662706138, −14.31389737379146, −13.82460674701587, −13.19362219118236, −12.87718261312465, −11.91552869195590, −11.48010270724289, −11.13752759806509, −10.10398464440020, −9.883006482155254, −9.000387321111059, −8.435379147245880, −8.077141690557460, −7.387991789450450, −6.771048465963531, −6.031653051633737, −5.164366516963955, −4.783477981961983, −3.854292051729348, −3.034283232679340, −2.695008256878966, −1.738127860764370, −0.4692855450168438, 0.4692855450168438, 1.738127860764370, 2.695008256878966, 3.034283232679340, 3.854292051729348, 4.783477981961983, 5.164366516963955, 6.031653051633737, 6.771048465963531, 7.387991789450450, 8.077141690557460, 8.435379147245880, 9.000387321111059, 9.883006482155254, 10.10398464440020, 11.13752759806509, 11.48010270724289, 11.91552869195590, 12.87718261312465, 13.19362219118236, 13.82460674701587, 14.31389737379146, 14.92591662706138, 15.40391328411178, 16.00666505856340

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