Properties

Label 2-15680-1.1-c1-0-29
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  + 2·3-s + 5-s + 9-s − 3·11-s − 5·13-s + 2·15-s + 6·17-s + 19-s + 3·23-s + 25-s − 4·27-s + 6·29-s − 4·31-s − 6·33-s − 11·37-s − 10·39-s + 3·41-s + 10·43-s + 45-s + 3·47-s + 12·51-s − 3·53-s − 3·55-s + 2·57-s + 4·61-s − 5·65-s + 4·67-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.904·11-s − 1.38·13-s + 0.516·15-s + 1.45·17-s + 0.229·19-s + 0.625·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s − 0.718·31-s − 1.04·33-s − 1.80·37-s − 1.60·39-s + 0.468·41-s + 1.52·43-s + 0.149·45-s + 0.437·47-s + 1.68·51-s − 0.412·53-s − 0.404·55-s + 0.264·57-s + 0.512·61-s − 0.620·65-s + 0.488·67-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: $\chi_{15680} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.227162959\)
\(L(\frac12)\) \(\approx\) \(3.227162959\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.76003689309952, −15.48069531771611, −14.60261428116471, −14.33694963967747, −14.02190198350642, −13.29346086422694, −12.63079313569114, −12.33902991823320, −11.57730010237174, −10.69629243803660, −10.18372789092968, −9.743864626137117, −9.089649557671305, −8.635215556357789, −7.737592092603494, −7.599632024750028, −6.884531378818870, −5.874948229917616, −5.269791664060713, −4.802129852451746, −3.705949475671780, −3.080597245664328, −2.532089830864222, −1.902867228656956, −0.7078991891944851, 0.7078991891944851, 1.902867228656956, 2.532089830864222, 3.080597245664328, 3.705949475671780, 4.802129852451746, 5.269791664060713, 5.874948229917616, 6.884531378818870, 7.599632024750028, 7.737592092603494, 8.635215556357789, 9.089649557671305, 9.743864626137117, 10.18372789092968, 10.69629243803660, 11.57730010237174, 12.33902991823320, 12.63079313569114, 13.29346086422694, 14.02190198350642, 14.33694963967747, 14.60261428116471, 15.48069531771611, 15.76003689309952

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