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Properties

Label 2-15680-1.1-c1-0-26
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  − 3·3-s − 5-s + 6·9-s − 5·11-s − 3·13-s + 3·15-s + 17-s − 6·19-s − 6·23-s + 25-s − 9·27-s + 9·29-s − 4·31-s + 15·33-s − 2·37-s + 9·39-s + 4·41-s + 10·43-s − 6·45-s − 47-s − 3·51-s − 4·53-s + 5·55-s + 18·57-s + 8·59-s − 8·61-s + 3·65-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.447·5-s + 2·9-s − 1.50·11-s − 0.832·13-s + 0.774·15-s + 0.242·17-s − 1.37·19-s − 1.25·23-s + 1/5·25-s − 1.73·27-s + 1.67·29-s − 0.718·31-s + 2.61·33-s − 0.328·37-s + 1.44·39-s + 0.624·41-s + 1.52·43-s − 0.894·45-s − 0.145·47-s − 0.420·51-s − 0.549·53-s + 0.674·55-s + 2.38·57-s + 1.04·59-s − 1.02·61-s + 0.372·65-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 \( 1 + p T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−16.18846584282964, −15.81877887910536, −15.55389107663141, −14.63347003134816, −14.13965851772898, −13.18009384413753, −12.66440993464444, −12.38360892410397, −11.84123729077938, −11.14541267346631, −10.70635952918884, −10.19900534332525, −9.865904815670543, −8.754618380766203, −8.080713003207868, −7.454786557893393, −6.983064759573995, −6.025726988534502, −5.882488492201512, −4.874090222985194, −4.678614374286593, −3.882701555940653, −2.726721201355670, −1.976949649308402, −0.6841299297487954, 0, 0.6841299297487954, 1.976949649308402, 2.726721201355670, 3.882701555940653, 4.678614374286593, 4.874090222985194, 5.882488492201512, 6.025726988534502, 6.983064759573995, 7.454786557893393, 8.080713003207868, 8.754618380766203, 9.865904815670543, 10.19900534332525, 10.70635952918884, 11.14541267346631, 11.84123729077938, 12.38360892410397, 12.66440993464444, 13.18009384413753, 14.13965851772898, 14.63347003134816, 15.55389107663141, 15.81877887910536, 16.18846584282964

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