Properties

Label 2-15680-1.1-c1-0-15
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  − 3·3-s + 5-s + 6·9-s + 2·11-s − 3·15-s − 4·17-s + 6·19-s + 3·23-s + 25-s − 9·27-s − 9·29-s − 4·31-s − 6·33-s + 4·37-s − 7·41-s + 5·43-s + 6·45-s + 8·47-s + 12·51-s + 2·53-s + 2·55-s − 18·57-s − 10·59-s − 61-s + 9·67-s − 9·69-s + 2·71-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.447·5-s + 2·9-s + 0.603·11-s − 0.774·15-s − 0.970·17-s + 1.37·19-s + 0.625·23-s + 1/5·25-s − 1.73·27-s − 1.67·29-s − 0.718·31-s − 1.04·33-s + 0.657·37-s − 1.09·41-s + 0.762·43-s + 0.894·45-s + 1.16·47-s + 1.68·51-s + 0.274·53-s + 0.269·55-s − 2.38·57-s − 1.30·59-s − 0.128·61-s + 1.09·67-s − 1.08·69-s + 0.237·71-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: $\chi_{15680} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.161019715\)
\(L(\frac12)\) \(\approx\) \(1.161019715\)
\(L(\frac{3}{2})\) 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 \( 1 + p T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−16.10313281490497, −15.61831776293253, −15.01526956190044, −14.35231221015476, −13.49389795903986, −13.25827893435333, −12.41480743012073, −12.11341821365255, −11.27916444106050, −11.14322993779387, −10.55773282249377, −9.739231785632779, −9.353654679235816, −8.746367455555247, −7.492623597494513, −7.260079624027713, −6.453830749698008, −6.020994275386899, −5.353481420684285, −4.945901921130256, −4.139356461514142, −3.413288875190246, −2.195119768561681, −1.370163194962922, −0.5580680842606200, 0.5580680842606200, 1.370163194962922, 2.195119768561681, 3.413288875190246, 4.139356461514142, 4.945901921130256, 5.353481420684285, 6.020994275386899, 6.453830749698008, 7.260079624027713, 7.492623597494513, 8.746367455555247, 9.353654679235816, 9.739231785632779, 10.55773282249377, 11.14322993779387, 11.27916444106050, 12.11341821365255, 12.41480743012073, 13.25827893435333, 13.49389795903986, 14.35231221015476, 15.01526956190044, 15.61831776293253, 16.10313281490497

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