Properties

Label 2-15680-1.1-c1-0-0
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  − 3·3-s − 5-s + 6·9-s + 11-s − 13-s + 3·15-s + 3·17-s − 8·19-s − 4·23-s + 25-s − 9·27-s − 3·29-s − 6·31-s − 3·33-s + 8·37-s + 3·39-s − 10·41-s − 12·43-s − 6·45-s + 3·47-s − 9·51-s − 12·53-s − 55-s + 24·57-s + 2·61-s + 65-s + 4·67-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.447·5-s + 2·9-s + 0.301·11-s − 0.277·13-s + 0.774·15-s + 0.727·17-s − 1.83·19-s − 0.834·23-s + 1/5·25-s − 1.73·27-s − 0.557·29-s − 1.07·31-s − 0.522·33-s + 1.31·37-s + 0.480·39-s − 1.56·41-s − 1.82·43-s − 0.894·45-s + 0.437·47-s − 1.26·51-s − 1.64·53-s − 0.134·55-s + 3.17·57-s + 0.256·61-s + 0.124·65-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

\(L(1)\) \(\approx\) \(0.3177658826\)
\(L(\frac12)\) \(\approx\) \(0.3177658826\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−16.21174339159290, −15.53934114093457, −14.92910042900086, −14.58558106411344, −13.60760627601569, −12.94832385411374, −12.48538581900562, −12.04315281551924, −11.53715093132784, −10.89330468699863, −10.64457124441107, −9.832088098310839, −9.422147585273635, −8.319085428061963, −7.930278202314037, −7.045894264596435, −6.502754758698321, −6.133655058136076, −5.256269123481689, −4.895346476650858, −4.043014307859052, −3.576485636939407, −2.199709331641168, −1.408040834956991, −0.2796364050615939, 0.2796364050615939, 1.408040834956991, 2.199709331641168, 3.576485636939407, 4.043014307859052, 4.895346476650858, 5.256269123481689, 6.133655058136076, 6.502754758698321, 7.045894264596435, 7.930278202314037, 8.319085428061963, 9.422147585273635, 9.832088098310839, 10.64457124441107, 10.89330468699863, 11.53715093132784, 12.04315281551924, 12.48538581900562, 12.94832385411374, 13.60760627601569, 14.58558106411344, 14.92910042900086, 15.53934114093457, 16.21174339159290

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