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Properties

Label 2-63-63.61-c2-0-13
Degree $2$
Conductor $63$
Sign $-0.924 + 0.381i$
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  + (−0.840 − 1.45i)2-s + (−1.62 − 2.52i)3-s + (0.585 − 1.01i)4-s + 2.34i·5-s + (−2.31 + 4.48i)6-s + (−3.93 − 5.78i)7-s − 8.69·8-s + (−3.73 + 8.18i)9-s + (3.41 − 1.97i)10-s − 6.20·11-s + (−3.50 + 0.167i)12-s + (21.3 − 12.3i)13-s + (−5.12 + 10.6i)14-s + (5.91 − 3.80i)15-s + (4.97 + 8.61i)16-s + (19.5 − 11.2i)17-s + ⋯
L(s)  = 1  + (−0.420 − 0.728i)2-s + (−0.540 − 0.841i)3-s + (0.146 − 0.253i)4-s + 0.468i·5-s + (−0.385 + 0.747i)6-s + (−0.562 − 0.827i)7-s − 1.08·8-s + (−0.415 + 0.909i)9-s + (0.341 − 0.197i)10-s − 0.564·11-s + (−0.292 + 0.0139i)12-s + (1.64 − 0.948i)13-s + (−0.366 + 0.757i)14-s + (0.394 − 0.253i)15-s + (0.310 + 0.538i)16-s + (1.14 − 0.662i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.381i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.924 + 0.381i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ -0.924 + 0.381i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.143761 - 0.725975i\)
\(L(\frac12)\) \(\approx\) \(0.143761 - 0.725975i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.62 + 2.52i)T \)
7 \( 1 + (3.93 + 5.78i)T \)
good2 \( 1 + (0.840 + 1.45i)T + (-2 + 3.46i)T^{2} \)
5 \( 1 - 2.34iT - 25T^{2} \)
11 \( 1 + 6.20T + 121T^{2} \)
13 \( 1 + (-21.3 + 12.3i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-19.5 + 11.2i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (9.34 + 5.39i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + 8.54T + 529T^{2} \)
29 \( 1 + (-16.1 + 27.9i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-1.44 - 0.833i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (19.7 - 34.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (27.9 - 16.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (2.15 - 3.73i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-42.0 + 24.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (1.04 + 1.81i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-91.7 - 52.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (20.0 - 11.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-1.29 + 2.24i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 66.6T + 5.04e3T^{2} \)
73 \( 1 + (-18.4 + 10.6i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-51.5 - 89.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (10.0 + 5.80i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (12.0 + 6.93i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (13.0 + 7.54i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−13.89354279539675408914156458267, −13.00705583344768856354481072879, −11.78830616867558922085472967600, −10.71538796807336084003395189244, −10.17870658943654513413621155559, −8.272730578196727304912230522820, −6.82900992215145405865536078400, −5.75595847309413337872808989244, −3.01938336937109282534644160033, −0.868770893512026453407523140405, 3.57301745582915755406176451562, 5.57312106402715644391871202135, 6.50924974475928480398944676426, 8.445225417174227277147589367089, 9.096704151987272801233137407012, 10.55189286163180638149902956914, 11.90019204968747745860252522727, 12.72605783038464755236279205136, 14.54527374026805120486063797275, 15.75816931332958893703032485421

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