Properties

Label 2-2e3-1.1-c15-0-0
Degree $2$
Conductor $8$
Sign $1$
Analytic cond. $11.4154$
Root an. cond. $3.37868$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.91e3·3-s − 2.45e5·5-s − 3.75e5·7-s + 3.34e7·9-s − 7.40e7·11-s + 2.38e8·13-s + 1.69e9·15-s + 7.12e8·17-s − 1.63e9·19-s + 2.59e9·21-s − 1.84e10·23-s + 2.98e10·25-s − 1.31e11·27-s − 7.74e10·29-s + 1.32e11·31-s + 5.11e11·33-s + 9.22e10·35-s + 1.16e11·37-s − 1.64e12·39-s + 2.32e12·41-s + 2.08e12·43-s − 8.20e12·45-s + 2.33e12·47-s − 4.60e12·49-s − 4.92e12·51-s + 6.30e12·53-s + 1.81e13·55-s + ⋯
L(s)  = 1  − 1.82·3-s − 1.40·5-s − 0.172·7-s + 2.32·9-s − 1.14·11-s + 1.05·13-s + 2.56·15-s + 0.421·17-s − 0.418·19-s + 0.314·21-s − 1.12·23-s + 0.976·25-s − 2.42·27-s − 0.833·29-s + 0.868·31-s + 2.08·33-s + 0.242·35-s + 0.202·37-s − 1.92·39-s + 1.86·41-s + 1.17·43-s − 3.27·45-s + 0.673·47-s − 0.970·49-s − 0.768·51-s + 0.737·53-s + 1.60·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $1$
Analytic conductor: \(11.4154\)
Root analytic conductor: \(3.37868\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :15/2),\ 1)\)

Particular Values

\(L(8)\) \(\approx\) \(0.4575655979\)
\(L(\frac12)\) \(\approx\) \(0.4575655979\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 6.91e3T + 1.43e7T^{2} \)
5 \( 1 + 2.45e5T + 3.05e10T^{2} \)
7 \( 1 + 3.75e5T + 4.74e12T^{2} \)
11 \( 1 + 7.40e7T + 4.17e15T^{2} \)
13 \( 1 - 2.38e8T + 5.11e16T^{2} \)
17 \( 1 - 7.12e8T + 2.86e18T^{2} \)
19 \( 1 + 1.63e9T + 1.51e19T^{2} \)
23 \( 1 + 1.84e10T + 2.66e20T^{2} \)
29 \( 1 + 7.74e10T + 8.62e21T^{2} \)
31 \( 1 - 1.32e11T + 2.34e22T^{2} \)
37 \( 1 - 1.16e11T + 3.33e23T^{2} \)
41 \( 1 - 2.32e12T + 1.55e24T^{2} \)
43 \( 1 - 2.08e12T + 3.17e24T^{2} \)
47 \( 1 - 2.33e12T + 1.20e25T^{2} \)
53 \( 1 - 6.30e12T + 7.31e25T^{2} \)
59 \( 1 - 2.90e12T + 3.65e26T^{2} \)
61 \( 1 - 2.66e12T + 6.02e26T^{2} \)
67 \( 1 - 1.67e12T + 2.46e27T^{2} \)
71 \( 1 + 8.11e13T + 5.87e27T^{2} \)
73 \( 1 + 9.97e13T + 8.90e27T^{2} \)
79 \( 1 - 7.04e13T + 2.91e28T^{2} \)
83 \( 1 - 2.99e14T + 6.11e28T^{2} \)
89 \( 1 + 6.60e14T + 1.74e29T^{2} \)
97 \( 1 - 1.46e15T + 6.33e29T^{2} \)
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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

−17.93141819570888901536804707999, −16.28672563384899712417247302567, −15.66908827177854488616155138602, −12.77475607150245383652151371922, −11.59420983772793419292327645457, −10.57462376958295016153351411067, −7.67793198394861137419437457875, −5.90039996848995902934037329616, −4.19861867695133090937414233451, −0.56824712043044151201863432086, 0.56824712043044151201863432086, 4.19861867695133090937414233451, 5.90039996848995902934037329616, 7.67793198394861137419437457875, 10.57462376958295016153351411067, 11.59420983772793419292327645457, 12.77475607150245383652151371922, 15.66908827177854488616155138602, 16.28672563384899712417247302567, 17.93141819570888901536804707999

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