Properties

Label 2-15680-1.1-c1-0-32
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  − 3-s + 5-s − 2·9-s + 5·11-s + 5·13-s − 15-s + 5·17-s − 8·23-s + 25-s + 5·27-s + 29-s + 2·31-s − 5·33-s − 4·37-s − 5·39-s − 2·41-s + 4·43-s − 2·45-s + 13·47-s − 5·51-s + 8·53-s + 5·55-s − 4·59-s − 2·61-s + 5·65-s − 8·67-s + 8·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 2/3·9-s + 1.50·11-s + 1.38·13-s − 0.258·15-s + 1.21·17-s − 1.66·23-s + 1/5·25-s + 0.962·27-s + 0.185·29-s + 0.359·31-s − 0.870·33-s − 0.657·37-s − 0.800·39-s − 0.312·41-s + 0.609·43-s − 0.298·45-s + 1.89·47-s − 0.700·51-s + 1.09·53-s + 0.674·55-s − 0.520·59-s − 0.256·61-s + 0.620·65-s − 0.977·67-s + 0.963·69-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: \(0\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.360360577\)
\(L(\frac12)\) \(\approx\) \(2.360360577\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−16.00426874593772, −15.65172433135851, −14.70343293166819, −14.20468129687836, −13.92261147817658, −13.37382629156215, −12.31678128754055, −12.10110929224962, −11.61510983595532, −10.93530688441332, −10.38076787577355, −9.842573727259171, −9.010247214269116, −8.699986853395030, −7.994233336578648, −7.188387696387334, −6.392623278800964, −5.942498959634908, −5.696356304781804, −4.697519002605806, −3.837822937511821, −3.454308653590802, −2.358454679108069, −1.418207216477782, −0.7579354839384430, 0.7579354839384430, 1.418207216477782, 2.358454679108069, 3.454308653590802, 3.837822937511821, 4.697519002605806, 5.696356304781804, 5.942498959634908, 6.392623278800964, 7.188387696387334, 7.994233336578648, 8.699986853395030, 9.010247214269116, 9.842573727259171, 10.38076787577355, 10.93530688441332, 11.61510983595532, 12.10110929224962, 12.31678128754055, 13.37382629156215, 13.92261147817658, 14.20468129687836, 14.70343293166819, 15.65172433135851, 16.00426874593772

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