Properties

Label 2-63-63.32-c2-0-4
Degree $2$
Conductor $63$
Sign $-0.177 + 0.984i$
Analytic cond. $1.71662$
Root an. cond. $1.31020$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.79 − 1.61i)2-s + (−0.476 + 2.96i)3-s + (3.19 + 5.53i)4-s − 5.53i·5-s + (6.10 − 7.50i)6-s + (3.31 − 6.16i)7-s − 7.70i·8-s + (−8.54 − 2.82i)9-s + (−8.91 + 15.4i)10-s − 17.6i·11-s + (−17.9 + 6.82i)12-s + (2.03 − 3.52i)13-s + (−19.2 + 11.8i)14-s + (16.3 + 2.63i)15-s + (0.357 − 0.619i)16-s + (14.3 + 8.27i)17-s + ⋯
L(s)  = 1  + (−1.39 − 0.805i)2-s + (−0.158 + 0.987i)3-s + (0.798 + 1.38i)4-s − 1.10i·5-s + (1.01 − 1.25i)6-s + (0.474 − 0.880i)7-s − 0.963i·8-s + (−0.949 − 0.313i)9-s + (−0.891 + 1.54i)10-s − 1.60i·11-s + (−1.49 + 0.568i)12-s + (0.156 − 0.271i)13-s + (−1.37 + 0.846i)14-s + (1.09 + 0.175i)15-s + (0.0223 − 0.0387i)16-s + (0.843 + 0.486i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.177 + 0.984i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ -0.177 + 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.346860 - 0.414981i\)
\(L(\frac12)\) \(\approx\) \(0.346860 - 0.414981i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.476 - 2.96i)T \)
7 \( 1 + (-3.31 + 6.16i)T \)
good2 \( 1 + (2.79 + 1.61i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + 5.53iT - 25T^{2} \)
11 \( 1 + 17.6iT - 121T^{2} \)
13 \( 1 + (-2.03 + 3.52i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-14.3 - 8.27i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (3.92 + 6.79i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 - 10.0iT - 529T^{2} \)
29 \( 1 + (39.9 - 23.0i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (14.8 + 25.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-15.5 - 27.0i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-27.8 - 16.0i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-3.35 - 5.80i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (14.2 + 8.22i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-32.5 - 18.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-82.3 + 47.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-36.8 + 63.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-6.09 - 10.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 20.0iT - 5.04e3T^{2} \)
73 \( 1 + (11.4 - 19.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (69.4 - 120. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-13.6 + 7.90i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-46.9 + 27.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-86.1 - 149. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−14.49881202639191590788923817796, −13.04712728202681102399334430472, −11.46955676986593594377501911929, −10.91483559391297929712211496873, −9.792419383795840433437263907178, −8.770214903747921264962309731328, −8.006217542912487727683078770614, −5.42158653487971202305465683407, −3.61882994433839337791912056303, −0.834096850410470215673033276020, 2.07646803732197152367700014677, 5.79878510425968219247143107916, 7.04232099017746225112755411446, 7.64273852516176426492555901325, 8.979845694178800517491909476885, 10.22710665491133484083102411982, 11.47832268517965159676787090494, 12.64218551765082818359863443330, 14.54124080118541266881322731546, 14.93715742210292106652629853340

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