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Properties

Label 2-15680-1.1-c1-0-51
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 2·9-s + 3·11-s − 13-s + 15-s − 5·17-s − 6·19-s + 25-s + 5·27-s + 5·29-s − 2·31-s − 3·33-s + 4·37-s + 39-s − 2·41-s + 10·43-s + 2·45-s + 9·47-s + 5·51-s − 6·53-s − 3·55-s + 6·57-s − 6·59-s + 12·61-s + 65-s − 2·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.904·11-s − 0.277·13-s + 0.258·15-s − 1.21·17-s − 1.37·19-s + 1/5·25-s + 0.962·27-s + 0.928·29-s − 0.359·31-s − 0.522·33-s + 0.657·37-s + 0.160·39-s − 0.312·41-s + 1.52·43-s + 0.298·45-s + 1.31·47-s + 0.700·51-s − 0.824·53-s − 0.404·55-s + 0.794·57-s − 0.781·59-s + 1.53·61-s + 0.124·65-s − 0.244·67-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−16.26915966367854, −15.79778185867017, −15.14321347797224, −14.62235626543621, −14.16441921773535, −13.51419893285825, −12.70204511330969, −12.39847613731656, −11.66698629651409, −11.29574618531519, −10.74637683497733, −10.25035192269419, −9.260419061274222, −8.839838640999551, −8.379929753374702, −7.532880943023111, −6.864245193616678, −6.261817671816554, −5.898779760669487, −4.856192615940822, −4.403737694432140, −3.769334643274700, −2.760046658424459, −2.112338427470954, −0.9087992454753121, 0, 0.9087992454753121, 2.112338427470954, 2.760046658424459, 3.769334643274700, 4.403737694432140, 4.856192615940822, 5.898779760669487, 6.261817671816554, 6.864245193616678, 7.532880943023111, 8.379929753374702, 8.839838640999551, 9.260419061274222, 10.25035192269419, 10.74637683497733, 11.29574618531519, 11.66698629651409, 12.39847613731656, 12.70204511330969, 13.51419893285825, 14.16441921773535, 14.62235626543621, 15.14321347797224, 15.79778185867017, 16.26915966367854

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