Properties

Label 2-9680-1.1-c1-0-83
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  + 1.29·3-s + 5-s + 0.227·7-s − 1.32·9-s − 3.34·13-s + 1.29·15-s + 6.04·17-s + 2.22·19-s + 0.294·21-s + 3.47·23-s + 25-s − 5.59·27-s + 0.0307·29-s − 2.21·31-s + 0.227·35-s + 8.74·37-s − 4.33·39-s + 10.2·41-s + 3.74·43-s − 1.32·45-s + 5.31·47-s − 6.94·49-s + 7.83·51-s − 10.2·53-s + 2.87·57-s − 8.34·59-s − 2.24·61-s + ⋯
L(s)  = 1  + 0.747·3-s + 0.447·5-s + 0.0860·7-s − 0.441·9-s − 0.927·13-s + 0.334·15-s + 1.46·17-s + 0.509·19-s + 0.0643·21-s + 0.725·23-s + 0.200·25-s − 1.07·27-s + 0.00571·29-s − 0.398·31-s + 0.0385·35-s + 1.43·37-s − 0.693·39-s + 1.59·41-s + 0.570·43-s − 0.197·45-s + 0.775·47-s − 0.992·49-s + 1.09·51-s − 1.40·53-s + 0.381·57-s − 1.08·59-s − 0.287·61-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

\(L(1)\) \(\approx\) \(2.974867034\)
\(L(\frac12)\) \(\approx\) \(2.974867034\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.29T + 3T^{2} \)
7 \( 1 - 0.227T + 7T^{2} \)
13 \( 1 + 3.34T + 13T^{2} \)
17 \( 1 - 6.04T + 17T^{2} \)
19 \( 1 - 2.22T + 19T^{2} \)
23 \( 1 - 3.47T + 23T^{2} \)
29 \( 1 - 0.0307T + 29T^{2} \)
31 \( 1 + 2.21T + 31T^{2} \)
37 \( 1 - 8.74T + 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 - 3.74T + 43T^{2} \)
47 \( 1 - 5.31T + 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 + 8.34T + 59T^{2} \)
61 \( 1 + 2.24T + 61T^{2} \)
67 \( 1 + 3.47T + 67T^{2} \)
71 \( 1 - 15.0T + 71T^{2} \)
73 \( 1 + 3.66T + 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + 1.52T + 83T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 + 8.33T + 97T^{2} \)
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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

−7.75860504551836914331829486264, −7.24132280083312111276597201712, −6.21344088630855468157319133617, −5.63554933880577019769349259713, −4.99632674764594833206379205178, −4.13939129280876267233560263869, −3.11554385173556722434071363036, −2.79818131626089890876378031388, −1.85103258562836529007535936596, −0.794457921655297585609203072079, 0.794457921655297585609203072079, 1.85103258562836529007535936596, 2.79818131626089890876378031388, 3.11554385173556722434071363036, 4.13939129280876267233560263869, 4.99632674764594833206379205178, 5.63554933880577019769349259713, 6.21344088630855468157319133617, 7.24132280083312111276597201712, 7.75860504551836914331829486264

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