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Properties

Label 2-15680-1.1-c1-0-58
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  − 2·3-s + 5-s + 9-s − 11-s + 3·13-s − 2·15-s − 2·17-s − 5·19-s − 7·23-s + 25-s + 4·27-s + 6·29-s − 4·31-s + 2·33-s + 5·37-s − 6·39-s − 5·41-s + 6·43-s + 45-s + 9·47-s + 4·51-s − 11·53-s − 55-s + 10·57-s + 8·59-s + 12·61-s + 3·65-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.832·13-s − 0.516·15-s − 0.485·17-s − 1.14·19-s − 1.45·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s − 0.718·31-s + 0.348·33-s + 0.821·37-s − 0.960·39-s − 0.780·41-s + 0.914·43-s + 0.149·45-s + 1.31·47-s + 0.560·51-s − 1.51·53-s − 0.134·55-s + 1.32·57-s + 1.04·59-s + 1.53·61-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−16.29581620802700, −15.83926449327759, −15.39130689619518, −14.44486904297789, −14.12328799377865, −13.39918101944377, −12.82786522854748, −12.37651752785354, −11.74002994628161, −11.10790847455155, −10.77203406926588, −10.18712191680710, −9.631745846845147, −8.666133694857636, −8.418554827017384, −7.539258933817605, −6.637076332001437, −6.294518427728602, −5.771751519318582, −5.168360092669968, −4.381116625703600, −3.832089751886315, −2.687161892886496, −1.995612388817922, −0.9601065126787629, 0, 0.9601065126787629, 1.995612388817922, 2.687161892886496, 3.832089751886315, 4.381116625703600, 5.168360092669968, 5.771751519318582, 6.294518427728602, 6.637076332001437, 7.539258933817605, 8.418554827017384, 8.666133694857636, 9.631745846845147, 10.18712191680710, 10.77203406926588, 11.10790847455155, 11.74002994628161, 12.37651752785354, 12.82786522854748, 13.39918101944377, 14.12328799377865, 14.44486904297789, 15.39130689619518, 15.83926449327759, 16.29581620802700

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