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Properties

Label 2-15680-1.1-c1-0-17
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  − 5-s − 3·9-s + 4·11-s + 2·13-s − 6·17-s + 4·19-s + 8·23-s + 25-s − 6·29-s − 10·37-s − 10·41-s + 12·43-s + 3·45-s + 6·53-s − 4·55-s − 12·59-s + 2·61-s − 2·65-s + 4·67-s + 12·71-s + 10·73-s + 4·79-s + 9·81-s + 6·85-s − 18·89-s − 4·95-s + 10·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 9-s + 1.20·11-s + 0.554·13-s − 1.45·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s − 1.64·37-s − 1.56·41-s + 1.82·43-s + 0.447·45-s + 0.824·53-s − 0.539·55-s − 1.56·59-s + 0.256·61-s − 0.248·65-s + 0.488·67-s + 1.42·71-s + 1.17·73-s + 0.450·79-s + 81-s + 0.650·85-s − 1.90·89-s − 0.410·95-s + 1.01·97-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.694612791\)
\(L(\frac12)\) \(\approx\) \(1.694612791\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.87621744653152, −15.38186081978821, −15.02797321741189, −14.21067571465158, −13.88204341512059, −13.31394141999950, −12.56541768109057, −12.02044271690473, −11.31130055670285, −11.16991700238121, −10.56412410974126, −9.485927330358383, −9.009259464108354, −8.751985361578348, −8.005709412117634, −7.090646604439739, −6.811774167684425, −6.042120138839785, −5.310148404783287, −4.722937152220138, −3.748283059628746, −3.430070391599694, −2.494581346418890, −1.549122329974061, −0.5772213388296106, 0.5772213388296106, 1.549122329974061, 2.494581346418890, 3.430070391599694, 3.748283059628746, 4.722937152220138, 5.310148404783287, 6.042120138839785, 6.811774167684425, 7.090646604439739, 8.005709412117634, 8.751985361578348, 9.009259464108354, 9.485927330358383, 10.56412410974126, 11.16991700238121, 11.31130055670285, 12.02044271690473, 12.56541768109057, 13.31394141999950, 13.88204341512059, 14.21067571465158, 15.02797321741189, 15.38186081978821, 15.87621744653152

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