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Properties

Label 2-63-9.5-c2-0-1
Degree $2$
Conductor $63$
Sign $0.408 - 0.912i$
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  + (0.744 − 0.429i)2-s + (−2.65 + 1.40i)3-s + (−1.63 + 2.82i)4-s + (5.58 + 3.22i)5-s + (−1.36 + 2.18i)6-s + (1.32 + 2.29i)7-s + 6.24i·8-s + (5.04 − 7.45i)9-s + 5.54·10-s + (−11.7 + 6.80i)11-s + (0.351 − 9.77i)12-s + (9.13 − 15.8i)13-s + (1.96 + 1.13i)14-s + (−19.3 − 0.694i)15-s + (−3.83 − 6.64i)16-s − 4.81i·17-s + ⋯
L(s)  = 1  + (0.372 − 0.214i)2-s + (−0.883 + 0.468i)3-s + (−0.407 + 0.705i)4-s + (1.11 + 0.644i)5-s + (−0.228 + 0.364i)6-s + (0.188 + 0.327i)7-s + 0.780i·8-s + (0.560 − 0.827i)9-s + 0.554·10-s + (−1.07 + 0.618i)11-s + (0.0292 − 0.814i)12-s + (0.702 − 1.21i)13-s + (0.140 + 0.0812i)14-s + (−1.28 − 0.0463i)15-s + (−0.239 − 0.415i)16-s − 0.283i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.408 - 0.912i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ 0.408 - 0.912i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.982727 + 0.636761i\)
\(L(\frac12)\) \(\approx\) \(0.982727 + 0.636761i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.65 - 1.40i)T \)
7 \( 1 + (-1.32 - 2.29i)T \)
good2 \( 1 + (-0.744 + 0.429i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (-5.58 - 3.22i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (11.7 - 6.80i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-9.13 + 15.8i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 4.81iT - 289T^{2} \)
19 \( 1 - 14.3T + 361T^{2} \)
23 \( 1 + (-21.4 - 12.4i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-3.66 + 2.11i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-28.8 + 50.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 27.0T + 1.36e3T^{2} \)
41 \( 1 + (-42.9 - 24.7i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-0.0645 - 0.111i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (49.3 - 28.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 5.32iT - 2.80e3T^{2} \)
59 \( 1 + (23.1 + 13.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (14.6 + 25.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (47.5 - 82.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 94.1iT - 5.04e3T^{2} \)
73 \( 1 - 5.12T + 5.32e3T^{2} \)
79 \( 1 + (73.3 + 126. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-35.8 + 20.6i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 25.7iT - 7.92e3T^{2} \)
97 \( 1 + (-6.86 - 11.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−14.94958745328330802352982440369, −13.51739407869945721702877249658, −12.83022444853507014154177273898, −11.53387738031994596954332228215, −10.47028169748590492650306244385, −9.456198875170213659342179897703, −7.69843482364783255164187070510, −5.92387540721010447953039992475, −4.93864943593780893785702244622, −2.95737678570871211769931101939, 1.30013087921298259997289060839, 4.81218993196432925283150006763, 5.66654756272941928863638490135, 6.76234656266885307711439945157, 8.742658802994709083516866803184, 10.08967941536482754927097278474, 11.02679411758524434419105255429, 12.62469869813485671094849379903, 13.57100133424789508228697565924, 14.00057774637515120025868872197

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