Properties

Label 2-15680-1.1-c1-0-22
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 + 5·13-s − 15-s − 3·17-s + 2·19-s − 6·23-s + 25-s − 5·27-s − 3·29-s + 4·31-s + 3·33-s − 2·37-s + 5·39-s + 12·41-s + 10·43-s + 2·45-s − 9·47-s − 3·51-s − 12·53-s − 3·55-s + 2·57-s + 8·61-s − 5·65-s + 4·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.904·11-s + 1.38·13-s − 0.258·15-s − 0.727·17-s + 0.458·19-s − 1.25·23-s + 1/5·25-s − 0.962·27-s − 0.557·29-s + 0.718·31-s + 0.522·33-s − 0.328·37-s + 0.800·39-s + 1.87·41-s + 1.52·43-s + 0.298·45-s − 1.31·47-s − 0.420·51-s − 1.64·53-s − 0.404·55-s + 0.264·57-s + 1.02·61-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

\(L(1)\) \(\approx\) \(2.357291486\)
\(L(\frac12)\) \(\approx\) \(2.357291486\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−15.82538760213753, −15.66859709008256, −14.65790251484416, −14.39069309019990, −13.87772190262290, −13.32614984307172, −12.69985652159001, −12.00600120293010, −11.33648899011574, −11.18986481339413, −10.39675382045061, −9.437906843006411, −9.180241753634545, −8.518377372019239, −7.990596266729919, −7.519846979485403, −6.426373144115531, −6.233645812990390, −5.429521252229629, −4.415183282102728, −3.867488581503400, −3.360732566399367, −2.491466301278878, −1.660150292276173, −0.6509978804692007, 0.6509978804692007, 1.660150292276173, 2.491466301278878, 3.360732566399367, 3.867488581503400, 4.415183282102728, 5.429521252229629, 6.233645812990390, 6.426373144115531, 7.519846979485403, 7.990596266729919, 8.518377372019239, 9.180241753634545, 9.437906843006411, 10.39675382045061, 11.18986481339413, 11.33648899011574, 12.00600120293010, 12.69985652159001, 13.32614984307172, 13.87772190262290, 14.39069309019990, 14.65790251484416, 15.66859709008256, 15.82538760213753

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