Properties

Label 2-9576-1.1-c1-0-40
Degree $2$
Conductor $9576$
Sign $1$
Analytic cond. $76.4647$
Root an. cond. $8.74441$
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  − 4.30·5-s + 7-s + 2.56·11-s + 2.81·13-s + 7.96·17-s + 19-s + 0.814·23-s + 13.5·25-s − 3.48·29-s + 3.38·31-s − 4.30·35-s + 3.75·37-s − 6.60·41-s − 10.2·47-s + 49-s − 0.511·53-s − 11.0·55-s + 13.9·59-s + 6·61-s − 12.1·65-s + 11.9·67-s − 5.02·71-s + 16.0·73-s + 2.56·77-s − 14.8·79-s + 2.01·83-s − 34.2·85-s + ⋯
L(s)  = 1  − 1.92·5-s + 0.377·7-s + 0.773·11-s + 0.780·13-s + 1.93·17-s + 0.229·19-s + 0.169·23-s + 2.70·25-s − 0.647·29-s + 0.607·31-s − 0.727·35-s + 0.616·37-s − 1.03·41-s − 1.49·47-s + 0.142·49-s − 0.0702·53-s − 1.48·55-s + 1.82·59-s + 0.768·61-s − 1.50·65-s + 1.46·67-s − 0.596·71-s + 1.87·73-s + 0.292·77-s − 1.66·79-s + 0.221·83-s − 3.71·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9576\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(76.4647\)
Root analytic conductor: \(8.74441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{9576} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9576,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.746483625\)
\(L(\frac12)\) \(\approx\) \(1.746483625\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
19 \( 1 - T \)
good5 \( 1 + 4.30T + 5T^{2} \)
11 \( 1 - 2.56T + 11T^{2} \)
13 \( 1 - 2.81T + 13T^{2} \)
17 \( 1 - 7.96T + 17T^{2} \)
23 \( 1 - 0.814T + 23T^{2} \)
29 \( 1 + 3.48T + 29T^{2} \)
31 \( 1 - 3.38T + 31T^{2} \)
37 \( 1 - 3.75T + 37T^{2} \)
41 \( 1 + 6.60T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 + 0.511T + 53T^{2} \)
59 \( 1 - 13.9T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 11.9T + 67T^{2} \)
71 \( 1 + 5.02T + 71T^{2} \)
73 \( 1 - 16.0T + 73T^{2} \)
79 \( 1 + 14.8T + 79T^{2} \)
83 \( 1 - 2.01T + 83T^{2} \)
89 \( 1 + 17.6T + 89T^{2} \)
97 \( 1 + 5.66T + 97T^{2} \)
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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

−7.79353157073495250244653670466, −7.11646391828886085916920668897, −6.52504118779308284189520414993, −5.50396228115328199146031433338, −4.87714277435141712968607812847, −3.90254779641570082860597331351, −3.68183160137058929057553483277, −2.91902880629184358501316294680, −1.41525172964452349384319530777, −0.70732303832711008265485724378, 0.70732303832711008265485724378, 1.41525172964452349384319530777, 2.91902880629184358501316294680, 3.68183160137058929057553483277, 3.90254779641570082860597331351, 4.87714277435141712968607812847, 5.50396228115328199146031433338, 6.52504118779308284189520414993, 7.11646391828886085916920668897, 7.79353157073495250244653670466

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