Properties

Label 2-15680-1.1-c1-0-10
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 + 11-s − 3·13-s + 2·15-s + 2·17-s − 5·19-s + 7·23-s + 25-s + 4·27-s + 6·29-s − 4·31-s − 2·33-s + 5·37-s + 6·39-s + 5·41-s − 6·43-s − 45-s + 9·47-s − 4·51-s − 11·53-s − 55-s + 10·57-s + 8·59-s − 12·61-s + 3·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s − 0.832·13-s + 0.516·15-s + 0.485·17-s − 1.14·19-s + 1.45·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s − 0.718·31-s − 0.348·33-s + 0.821·37-s + 0.960·39-s + 0.780·41-s − 0.914·43-s − 0.149·45-s + 1.31·47-s − 0.560·51-s − 1.51·53-s − 0.134·55-s + 1.32·57-s + 1.04·59-s − 1.53·61-s + 0.372·65-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

\(L(1)\) \(\approx\) \(0.8480810314\)
\(L(\frac12)\) \(\approx\) \(0.8480810314\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−16.27184313753234, −15.42323818537526, −14.89112234887189, −14.51820957392934, −13.78600542093917, −12.96434926013101, −12.43012867341892, −12.18931227666637, −11.40946985096198, −11.02780962441449, −10.53013419852260, −9.879045336863461, −9.133840470030979, −8.577171765788709, −7.819212886343284, −7.160990080659532, −6.611740477756106, −6.028600305559865, −5.327667831300753, −4.709359592807494, −4.241562035822716, −3.196258314866828, −2.522842716109068, −1.325417881017928, −0.4596674769608144, 0.4596674769608144, 1.325417881017928, 2.522842716109068, 3.196258314866828, 4.241562035822716, 4.709359592807494, 5.327667831300753, 6.028600305559865, 6.611740477756106, 7.160990080659532, 7.819212886343284, 8.577171765788709, 9.133840470030979, 9.879045336863461, 10.53013419852260, 11.02780962441449, 11.40946985096198, 12.18931227666637, 12.43012867341892, 12.96434926013101, 13.78600542093917, 14.51820957392934, 14.89112234887189, 15.42323818537526, 16.27184313753234

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