Properties

Label 2-15680-1.1-c1-0-100
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 2·9-s − 6·11-s − 2·13-s − 15-s − 6·17-s − 8·19-s + 3·23-s + 25-s + 5·27-s − 3·29-s + 2·31-s + 6·33-s − 8·37-s + 2·39-s − 3·41-s − 5·43-s − 2·45-s + 6·51-s − 12·53-s − 6·55-s + 8·57-s + 61-s − 2·65-s + 7·67-s − 3·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 2/3·9-s − 1.80·11-s − 0.554·13-s − 0.258·15-s − 1.45·17-s − 1.83·19-s + 0.625·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s + 0.359·31-s + 1.04·33-s − 1.31·37-s + 0.320·39-s − 0.468·41-s − 0.762·43-s − 0.298·45-s + 0.840·51-s − 1.64·53-s − 0.809·55-s + 1.05·57-s + 0.128·61-s − 0.248·65-s + 0.855·67-s − 0.361·69-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: \(2\)
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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
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

−16.69775070030947, −15.91132726373880, −15.50754979712856, −14.93902248554180, −14.41251431192746, −13.56408722567062, −13.21448370951907, −12.69968830186927, −12.18923622641552, −11.25457310008276, −10.94778623860852, −10.46841312573786, −9.935059191560529, −9.045114491564880, −8.513417961258326, −8.074986619008806, −7.129891891751458, −6.586613677193768, −6.030773844936076, −5.138853200300801, −5.016429827857342, −4.139956249686909, −2.965263293155532, −2.477479841937167, −1.756525017169309, 0, 0, 1.756525017169309, 2.477479841937167, 2.965263293155532, 4.139956249686909, 5.016429827857342, 5.138853200300801, 6.030773844936076, 6.586613677193768, 7.129891891751458, 8.074986619008806, 8.513417961258326, 9.045114491564880, 9.935059191560529, 10.46841312573786, 10.94778623860852, 11.25457310008276, 12.18923622641552, 12.69968830186927, 13.21448370951907, 13.56408722567062, 14.41251431192746, 14.93902248554180, 15.50754979712856, 15.91132726373880, 16.69775070030947

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