Properties

Label 2-15680-1.1-c1-0-6
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 + 5·13-s + 2·15-s − 6·17-s − 19-s + 3·23-s + 25-s + 4·27-s + 6·29-s + 4·31-s + 6·33-s − 11·37-s − 10·39-s − 3·41-s + 10·43-s − 45-s − 3·47-s + 12·51-s − 3·53-s + 3·55-s + 2·57-s − 4·61-s − 5·65-s + 4·67-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.904·11-s + 1.38·13-s + 0.516·15-s − 1.45·17-s − 0.229·19-s + 0.625·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s + 1.04·33-s − 1.80·37-s − 1.60·39-s − 0.468·41-s + 1.52·43-s − 0.149·45-s − 0.437·47-s + 1.68·51-s − 0.412·53-s + 0.404·55-s + 0.264·57-s − 0.512·61-s − 0.620·65-s + 0.488·67-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

\(L(1)\) \(\approx\) \(0.7289784317\)
\(L(\frac12)\) \(\approx\) \(0.7289784317\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.91887573196252, −15.54137631360141, −15.32109748411824, −14.06428521249056, −13.87197084786955, −13.01978133364957, −12.64571058645881, −12.02150581286189, −11.31869454073388, −11.00357942172026, −10.61368684458187, −9.995744231976327, −8.950148565191481, −8.538813112395714, −8.045284928110324, −6.993128694291125, −6.662301098866969, −6.017631555785470, −5.358525990233217, −4.742910979657633, −4.170615146155122, −3.244341680133022, −2.493162274336639, −1.351603657280222, −0.4197297729268202, 0.4197297729268202, 1.351603657280222, 2.493162274336639, 3.244341680133022, 4.170615146155122, 4.742910979657633, 5.358525990233217, 6.017631555785470, 6.662301098866969, 6.993128694291125, 8.045284928110324, 8.538813112395714, 8.950148565191481, 9.995744231976327, 10.61368684458187, 11.00357942172026, 11.31869454073388, 12.02150581286189, 12.64571058645881, 13.01978133364957, 13.87197084786955, 14.06428521249056, 15.32109748411824, 15.54137631360141, 15.91887573196252

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