Properties

Label 2-9680-1.1-c1-0-32
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.39·3-s − 5-s − 1.65·7-s − 1.05·9-s + 6.36·13-s + 1.39·15-s − 5.31·17-s + 4.36·19-s + 2.31·21-s + 5.57·23-s + 25-s + 5.65·27-s − 6.79·29-s − 0.259·31-s + 1.65·35-s − 0.791·37-s − 8.88·39-s − 2.74·41-s + 11.6·43-s + 1.05·45-s − 7.49·47-s − 4.25·49-s + 7.41·51-s − 1.84·53-s − 6.08·57-s + 7.15·59-s + 8.57·61-s + ⋯
L(s)  = 1  − 0.805·3-s − 0.447·5-s − 0.625·7-s − 0.350·9-s + 1.76·13-s + 0.360·15-s − 1.28·17-s + 1.00·19-s + 0.504·21-s + 1.16·23-s + 0.200·25-s + 1.08·27-s − 1.26·29-s − 0.0466·31-s + 0.279·35-s − 0.130·37-s − 1.42·39-s − 0.427·41-s + 1.77·43-s + 0.156·45-s − 1.09·47-s − 0.608·49-s + 1.03·51-s − 0.253·53-s − 0.806·57-s + 0.931·59-s + 1.09·61-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: \(0\)
Selberg data: \((2,\ 9680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9636269084\)
\(L(\frac12)\) \(\approx\) \(0.9636269084\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.39T + 3T^{2} \)
7 \( 1 + 1.65T + 7T^{2} \)
13 \( 1 - 6.36T + 13T^{2} \)
17 \( 1 + 5.31T + 17T^{2} \)
19 \( 1 - 4.36T + 19T^{2} \)
23 \( 1 - 5.57T + 23T^{2} \)
29 \( 1 + 6.79T + 29T^{2} \)
31 \( 1 + 0.259T + 31T^{2} \)
37 \( 1 + 0.791T + 37T^{2} \)
41 \( 1 + 2.74T + 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 + 7.49T + 47T^{2} \)
53 \( 1 + 1.84T + 53T^{2} \)
59 \( 1 - 7.15T + 59T^{2} \)
61 \( 1 - 8.57T + 61T^{2} \)
67 \( 1 + 15.6T + 67T^{2} \)
71 \( 1 + 1.05T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 2.27T + 79T^{2} \)
83 \( 1 - 7.15T + 83T^{2} \)
89 \( 1 + 8.31T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−7.54754024221834812180105851913, −6.85358002600755385382443969421, −6.29247377805664462273263214570, −5.71884120414030409485934180738, −5.04833142125580647618691462920, −4.16863742763785443656620995709, −3.45510226753073829710624914087, −2.79743155099485456501315371294, −1.47750738266244157610507631117, −0.51072267216545339265746537766, 0.51072267216545339265746537766, 1.47750738266244157610507631117, 2.79743155099485456501315371294, 3.45510226753073829710624914087, 4.16863742763785443656620995709, 5.04833142125580647618691462920, 5.71884120414030409485934180738, 6.29247377805664462273263214570, 6.85358002600755385382443969421, 7.54754024221834812180105851913

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