Properties

Label 2-15680-1.1-c1-0-75
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 − 2·11-s + 6·13-s − 3·15-s + 2·17-s + 9·23-s + 25-s − 9·27-s − 3·29-s − 2·31-s + 6·33-s − 8·37-s − 18·39-s + 5·41-s + 43-s + 6·45-s − 8·47-s − 6·51-s − 4·53-s − 2·55-s − 8·59-s − 7·61-s + 6·65-s − 3·67-s − 27·69-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.447·5-s + 2·9-s − 0.603·11-s + 1.66·13-s − 0.774·15-s + 0.485·17-s + 1.87·23-s + 1/5·25-s − 1.73·27-s − 0.557·29-s − 0.359·31-s + 1.04·33-s − 1.31·37-s − 2.88·39-s + 0.780·41-s + 0.152·43-s + 0.894·45-s − 1.16·47-s − 0.840·51-s − 0.549·53-s − 0.269·55-s − 1.04·59-s − 0.896·61-s + 0.744·65-s − 0.366·67-s − 3.25·69-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: \(1\)
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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
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

−16.53924276097805, −15.77857248371115, −15.45929705373703, −14.69731542667177, −13.90533249339385, −13.17456307650120, −13.01547542774781, −12.31966425352184, −11.72625330877742, −10.98154620917147, −10.82970110505596, −10.40383093810963, −9.462837645016167, −9.045919067362497, −8.145357139228173, −7.440537656264192, −6.682732071460288, −6.327731468127015, −5.553588370459019, −5.286907398886290, −4.584172422430412, −3.694621923759365, −2.935600884949372, −1.587870410305655, −1.111994902792796, 0, 1.111994902792796, 1.587870410305655, 2.935600884949372, 3.694621923759365, 4.584172422430412, 5.286907398886290, 5.553588370459019, 6.327731468127015, 6.682732071460288, 7.440537656264192, 8.145357139228173, 9.045919067362497, 9.462837645016167, 10.40383093810963, 10.82970110505596, 10.98154620917147, 11.72625330877742, 12.31966425352184, 13.01547542774781, 13.17456307650120, 13.90533249339385, 14.69731542667177, 15.45929705373703, 15.77857248371115, 16.53924276097805

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