Properties

Label 2-15680-1.1-c1-0-5
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 − 2·11-s − 15-s − 4·17-s − 2·19-s + 23-s + 25-s − 5·27-s − 9·29-s − 4·31-s − 2·33-s − 4·37-s − 41-s − 9·43-s + 2·45-s − 4·51-s + 10·53-s + 2·55-s − 2·57-s − 10·59-s + 9·61-s − 5·67-s + 69-s + 14·71-s − 12·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.603·11-s − 0.258·15-s − 0.970·17-s − 0.458·19-s + 0.208·23-s + 1/5·25-s − 0.962·27-s − 1.67·29-s − 0.718·31-s − 0.348·33-s − 0.657·37-s − 0.156·41-s − 1.37·43-s + 0.298·45-s − 0.560·51-s + 1.37·53-s + 0.269·55-s − 0.264·57-s − 1.30·59-s + 1.15·61-s − 0.610·67-s + 0.120·69-s + 1.66·71-s − 1.40·73-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

\(L(1)\) \(\approx\) \(1.008633865\)
\(L(\frac12)\) \(\approx\) \(1.008633865\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 9 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−15.87171452686465, −15.32564327550433, −14.88165597049428, −14.51799483779869, −13.65666695372938, −13.26363636671673, −12.85350715231784, −11.95687312362609, −11.55362363115422, −10.84019116463349, −10.53115689317315, −9.577836512869541, −9.001539228009039, −8.613166315952863, −7.941017399534951, −7.433789912218251, −6.753453429275240, −6.004215757823504, −5.278147731144270, −4.702702327939324, −3.691367312245575, −3.398985063657733, −2.351663529693428, −1.912868183478326, −0.3920367480778161, 0.3920367480778161, 1.912868183478326, 2.351663529693428, 3.398985063657733, 3.691367312245575, 4.702702327939324, 5.278147731144270, 6.004215757823504, 6.753453429275240, 7.433789912218251, 7.941017399534951, 8.613166315952863, 9.001539228009039, 9.577836512869541, 10.53115689317315, 10.84019116463349, 11.55362363115422, 11.95687312362609, 12.85350715231784, 13.26363636671673, 13.65666695372938, 14.51799483779869, 14.88165597049428, 15.32564327550433, 15.87171452686465

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