Properties

Label 2-9680-1.1-c1-0-153
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  + 3.14·3-s + 5-s + 3.14·7-s + 6.86·9-s − 7.00·13-s + 3.14·15-s + 5.14·17-s + 0.414·19-s + 9.86·21-s + 2.72·23-s + 25-s + 12.1·27-s + 5.86·29-s + 9.86·31-s + 3.14·35-s − 3.86·37-s − 22.0·39-s − 6.28·41-s − 6.72·43-s + 6.86·45-s + 9.00·47-s + 2.86·49-s + 16.1·51-s − 3.86·53-s + 1.30·57-s − 1.45·59-s + 10.6·61-s + ⋯
L(s)  = 1  + 1.81·3-s + 0.447·5-s + 1.18·7-s + 2.28·9-s − 1.94·13-s + 0.811·15-s + 1.24·17-s + 0.0951·19-s + 2.15·21-s + 0.568·23-s + 0.200·25-s + 2.33·27-s + 1.08·29-s + 1.77·31-s + 0.530·35-s − 0.635·37-s − 3.52·39-s − 0.981·41-s − 1.02·43-s + 1.02·45-s + 1.31·47-s + 0.409·49-s + 2.26·51-s − 0.531·53-s + 0.172·57-s − 0.189·59-s + 1.36·61-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

\(L(1)\) \(\approx\) \(5.808473946\)
\(L(\frac12)\) \(\approx\) \(5.808473946\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.14T + 3T^{2} \)
7 \( 1 - 3.14T + 7T^{2} \)
13 \( 1 + 7.00T + 13T^{2} \)
17 \( 1 - 5.14T + 17T^{2} \)
19 \( 1 - 0.414T + 19T^{2} \)
23 \( 1 - 2.72T + 23T^{2} \)
29 \( 1 - 5.86T + 29T^{2} \)
31 \( 1 - 9.86T + 31T^{2} \)
37 \( 1 + 3.86T + 37T^{2} \)
41 \( 1 + 6.28T + 41T^{2} \)
43 \( 1 + 6.72T + 43T^{2} \)
47 \( 1 - 9.00T + 47T^{2} \)
53 \( 1 + 3.86T + 53T^{2} \)
59 \( 1 + 1.45T + 59T^{2} \)
61 \( 1 - 10.6T + 61T^{2} \)
67 \( 1 + 7.29T + 67T^{2} \)
71 \( 1 - 6.69T + 71T^{2} \)
73 \( 1 + 8.72T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 - 4.17T + 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−7.77345969231527185453873568768, −7.28653034230018761533490745268, −6.63128699194487440843466359479, −5.31440924819345913393596772571, −4.84718785782607107581296774015, −4.20497667000711147105680773082, −3.08079211108406218832315421688, −2.70057732770798090186206355022, −1.89853826540730011602315577343, −1.14766842624033893569676927141, 1.14766842624033893569676927141, 1.89853826540730011602315577343, 2.70057732770798090186206355022, 3.08079211108406218832315421688, 4.20497667000711147105680773082, 4.84718785782607107581296774015, 5.31440924819345913393596772571, 6.63128699194487440843466359479, 7.28653034230018761533490745268, 7.77345969231527185453873568768

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