Properties

Degree $2$
Conductor $63$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.35·2-s + 11.0·4-s + 8.71·5-s − 7·7-s + 13.0·8-s + 38.0·10-s − 43.5·11-s + 82·13-s − 30.5·14-s − 30.9·16-s − 78.4·17-s − 20·19-s + 95.8·20-s − 190.·22-s − 130.·23-s − 48.9·25-s + 357.·26-s − 77.0·28-s + 244.·29-s + 156·31-s − 239.·32-s − 342.·34-s − 61.0·35-s + 186·37-s − 87.1·38-s + 114.·40-s − 165.·41-s + ⋯
L(s)  = 1  + 1.54·2-s + 1.37·4-s + 0.779·5-s − 0.377·7-s + 0.577·8-s + 1.20·10-s − 1.19·11-s + 1.74·13-s − 0.582·14-s − 0.484·16-s − 1.11·17-s − 0.241·19-s + 1.07·20-s − 1.84·22-s − 1.18·23-s − 0.391·25-s + 2.69·26-s − 0.519·28-s + 1.56·29-s + 0.903·31-s − 1.32·32-s − 1.72·34-s − 0.294·35-s + 0.826·37-s − 0.372·38-s + 0.450·40-s − 0.630·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $1$
Motivic weight: \(3\)
Character: $\chi_{63} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.08667\)
\(L(\frac12)\) \(\approx\) \(3.08667\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + 7T \)
good2 \( 1 - 4.35T + 8T^{2} \)
5 \( 1 - 8.71T + 125T^{2} \)
11 \( 1 + 43.5T + 1.33e3T^{2} \)
13 \( 1 - 82T + 2.19e3T^{2} \)
17 \( 1 + 78.4T + 4.91e3T^{2} \)
19 \( 1 + 20T + 6.85e3T^{2} \)
23 \( 1 + 130.T + 1.21e4T^{2} \)
29 \( 1 - 244.T + 2.43e4T^{2} \)
31 \( 1 - 156T + 2.97e4T^{2} \)
37 \( 1 - 186T + 5.06e4T^{2} \)
41 \( 1 + 165.T + 6.89e4T^{2} \)
43 \( 1 - 164T + 7.95e4T^{2} \)
47 \( 1 - 470.T + 1.03e5T^{2} \)
53 \( 1 + 156.T + 1.48e5T^{2} \)
59 \( 1 - 156.T + 2.05e5T^{2} \)
61 \( 1 - 790T + 2.26e5T^{2} \)
67 \( 1 + 44T + 3.00e5T^{2} \)
71 \( 1 - 444.T + 3.57e5T^{2} \)
73 \( 1 - 126T + 3.89e5T^{2} \)
79 \( 1 + 712T + 4.93e5T^{2} \)
83 \( 1 + 1.46e3T + 5.71e5T^{2} \)
89 \( 1 + 1.45e3T + 7.04e5T^{2} \)
97 \( 1 - 798T + 9.12e5T^{2} \)
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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

−13.96785455537949929393881666455, −13.50640275149007135629872319687, −12.69989022722962218748552327117, −11.33541398420770267525710753026, −10.18051851977067388998539362168, −8.498283235177754701987777796728, −6.48476799588701545015873416061, −5.70392303032643907704309463931, −4.18295387428861661860873926800, −2.54156397125768629133131145321, 2.54156397125768629133131145321, 4.18295387428861661860873926800, 5.70392303032643907704309463931, 6.48476799588701545015873416061, 8.498283235177754701987777796728, 10.18051851977067388998539362168, 11.33541398420770267525710753026, 12.69989022722962218748552327117, 13.50640275149007135629872319687, 13.96785455537949929393881666455

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