Properties

Label 2-15680-1.1-c1-0-49
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 + 11-s + 3·13-s + 2·15-s − 2·17-s + 5·19-s + 7·23-s + 25-s − 4·27-s + 6·29-s + 4·31-s + 2·33-s + 5·37-s + 6·39-s − 5·41-s − 6·43-s + 45-s − 9·47-s − 4·51-s − 11·53-s + 55-s + 10·57-s − 8·59-s + 12·61-s + 3·65-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s + 0.832·13-s + 0.516·15-s − 0.485·17-s + 1.14·19-s + 1.45·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.348·33-s + 0.821·37-s + 0.960·39-s − 0.780·41-s − 0.914·43-s + 0.149·45-s − 1.31·47-s − 0.560·51-s − 1.51·53-s + 0.134·55-s + 1.32·57-s − 1.04·59-s + 1.53·61-s + 0.372·65-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

\(L(1)\) \(\approx\) \(4.403420426\)
\(L(\frac12)\) \(\approx\) \(4.403420426\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−15.86010969381131, −15.34899553110521, −14.84087212580193, −14.25890426454881, −13.76026239459204, −13.41120931718765, −12.90117907947253, −12.12428304473187, −11.34248642085959, −11.07619888192510, −10.09411441563103, −9.646504967755703, −9.101858358958230, −8.549939478512254, −8.105863892904474, −7.419942715245321, −6.534139038117521, −6.290619832313561, −5.141630060271992, −4.769207196533571, −3.644510737747622, −3.218408580144813, −2.579507232107602, −1.682661693049572, −0.8991582229878423, 0.8991582229878423, 1.682661693049572, 2.579507232107602, 3.218408580144813, 3.644510737747622, 4.769207196533571, 5.141630060271992, 6.290619832313561, 6.534139038117521, 7.419942715245321, 8.105863892904474, 8.549939478512254, 9.101858358958230, 9.646504967755703, 10.09411441563103, 11.07619888192510, 11.34248642085959, 12.12428304473187, 12.90117907947253, 13.41120931718765, 13.76026239459204, 14.25890426454881, 14.84087212580193, 15.34899553110521, 15.86010969381131

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