Properties

Label 2-9680-1.1-c1-0-172
Degree $2$
Conductor $9680$
Sign $-1$
Analytic cond. $77.2951$
Root an. cond. $8.79176$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 5-s + 3.46·7-s + 9-s − 2·15-s + 6.92·17-s − 6.92·19-s − 6.92·21-s − 6·23-s + 25-s + 4·27-s − 4·31-s + 3.46·35-s + 10·37-s − 6.92·41-s − 3.46·43-s + 45-s + 6·47-s + 4.99·49-s − 13.8·51-s − 6·53-s + 13.8·57-s − 6.92·61-s + 3.46·63-s − 10·67-s + 12·69-s + 6.92·73-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s + 1.30·7-s + 0.333·9-s − 0.516·15-s + 1.68·17-s − 1.58·19-s − 1.51·21-s − 1.25·23-s + 0.200·25-s + 0.769·27-s − 0.718·31-s + 0.585·35-s + 1.64·37-s − 1.08·41-s − 0.528·43-s + 0.149·45-s + 0.875·47-s + 0.714·49-s − 1.94·51-s − 0.824·53-s + 1.83·57-s − 0.887·61-s + 0.436·63-s − 1.22·67-s + 1.44·69-s + 0.810·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9680\)    =    \(2^{4} \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(77.2951\)
Root analytic conductor: \(8.79176\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9680,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 \)
good3 \( 1 + 2T + 3T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 6.92T + 17T^{2} \)
19 \( 1 + 6.92T + 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 + 3.46T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 6.92T + 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6.92T + 73T^{2} \)
79 \( 1 + 6.92T + 79T^{2} \)
83 \( 1 + 17.3T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−7.30257145429964989470127056321, −6.46481495347833655471505935935, −5.77020538277290503662835174098, −5.52003923703056519246466974515, −4.63751013167091288395795264502, −4.14712762057067364199079420065, −2.95283724871260692501086256254, −1.90207122957652895029283014155, −1.24669399941346977336697117269, 0, 1.24669399941346977336697117269, 1.90207122957652895029283014155, 2.95283724871260692501086256254, 4.14712762057067364199079420065, 4.63751013167091288395795264502, 5.52003923703056519246466974515, 5.77020538277290503662835174098, 6.46481495347833655471505935935, 7.30257145429964989470127056321

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