Properties

Label 2-15680-1.1-c1-0-1
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  − 2·3-s − 5-s + 9-s + 3·11-s + 13-s + 2·15-s − 6·17-s − 19-s − 9·23-s + 25-s + 4·27-s − 6·29-s − 8·31-s − 6·33-s + 7·37-s − 2·39-s + 3·41-s + 2·43-s − 45-s − 9·47-s + 12·51-s − 9·53-s − 3·55-s + 2·57-s − 8·61-s − 65-s + 8·67-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 0.516·15-s − 1.45·17-s − 0.229·19-s − 1.87·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s − 1.43·31-s − 1.04·33-s + 1.15·37-s − 0.320·39-s + 0.468·41-s + 0.304·43-s − 0.149·45-s − 1.31·47-s + 1.68·51-s − 1.23·53-s − 0.404·55-s + 0.264·57-s − 1.02·61-s − 0.124·65-s + 0.977·67-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

\(L(1)\) \(\approx\) \(0.3997057647\)
\(L(\frac12)\) \(\approx\) \(0.3997057647\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−16.16917055030531, −15.61229512057653, −14.84817153486539, −14.45994585898405, −13.75370014370163, −13.05107373778488, −12.54168784979543, −11.98063062207968, −11.39622501590575, −11.06515206966507, −10.67523446714432, −9.647527444903487, −9.292681749591571, −8.505245831673863, −7.898362053791192, −7.170595979024867, −6.402866844209883, −6.165516163958895, −5.455147481837667, −4.597210231615900, −4.119357871052981, −3.465587643336563, −2.272621949206861, −1.509013855139809, −0.2931181839683186, 0.2931181839683186, 1.509013855139809, 2.272621949206861, 3.465587643336563, 4.119357871052981, 4.597210231615900, 5.455147481837667, 6.165516163958895, 6.402866844209883, 7.170595979024867, 7.898362053791192, 8.505245831673863, 9.292681749591571, 9.647527444903487, 10.67523446714432, 11.06515206966507, 11.39622501590575, 11.98063062207968, 12.54168784979543, 13.05107373778488, 13.75370014370163, 14.45994585898405, 14.84817153486539, 15.61229512057653, 16.16917055030531

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