Properties

Label 2-15680-1.1-c1-0-18
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 + 2·13-s + 2·15-s + 6·17-s − 4·19-s + 6·23-s + 25-s + 4·27-s − 6·29-s + 4·31-s − 2·37-s − 4·39-s − 6·41-s + 10·43-s − 45-s + 6·47-s − 12·51-s + 6·53-s + 8·57-s + 12·59-s + 2·61-s − 2·65-s − 2·67-s − 12·69-s − 12·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.917·19-s + 1.25·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.640·39-s − 0.937·41-s + 1.52·43-s − 0.149·45-s + 0.875·47-s − 1.68·51-s + 0.824·53-s + 1.05·57-s + 1.56·59-s + 0.256·61-s − 0.248·65-s − 0.244·67-s − 1.44·69-s − 1.42·71-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

\(L(1)\) \(\approx\) \(1.215592430\)
\(L(\frac12)\) \(\approx\) \(1.215592430\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.13123074367083, −15.56777343889858, −14.77025266716560, −14.62209208444951, −13.63400844647978, −13.13238392843479, −12.46378778587444, −12.01737156040302, −11.56179442794275, −10.87159422377451, −10.59642056439151, −9.939909246597060, −9.060661584083742, −8.581213127948732, −7.847715038972807, −7.176709441408121, −6.636251296686564, −5.842648446183121, −5.498436186071081, −4.793998383172877, −4.017746404932287, −3.362938118646155, −2.471187497605137, −1.260410108030976, −0.5782744470046328, 0.5782744470046328, 1.260410108030976, 2.471187497605137, 3.362938118646155, 4.017746404932287, 4.793998383172877, 5.498436186071081, 5.842648446183121, 6.636251296686564, 7.176709441408121, 7.847715038972807, 8.581213127948732, 9.060661584083742, 9.939909246597060, 10.59642056439151, 10.87159422377451, 11.56179442794275, 12.01737156040302, 12.46378778587444, 13.13238392843479, 13.63400844647978, 14.62209208444951, 14.77025266716560, 15.56777343889858, 16.13123074367083

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