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Properties

Label 2-63-63.11-c2-0-10
Degree $2$
Conductor $63$
Sign $-0.277 + 0.960i$
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  − 0.513i·2-s + (−2.25 − 1.97i)3-s + 3.73·4-s + (−6.08 − 3.51i)5-s + (−1.01 + 1.16i)6-s + (1.23 − 6.88i)7-s − 3.97i·8-s + (1.21 + 8.91i)9-s + (−1.80 + 3.12i)10-s + (3.15 − 1.82i)11-s + (−8.44 − 7.37i)12-s + (−3.79 − 6.56i)13-s + (−3.53 − 0.636i)14-s + (6.81 + 19.9i)15-s + 12.9·16-s + (17.5 + 10.1i)17-s + ⋯
L(s)  = 1  − 0.256i·2-s + (−0.753 − 0.657i)3-s + 0.934·4-s + (−1.21 − 0.702i)5-s + (−0.168 + 0.193i)6-s + (0.177 − 0.984i)7-s − 0.496i·8-s + (0.134 + 0.990i)9-s + (−0.180 + 0.312i)10-s + (0.287 − 0.165i)11-s + (−0.703 − 0.614i)12-s + (−0.291 − 0.505i)13-s + (−0.252 − 0.0454i)14-s + (0.454 + 1.32i)15-s + 0.806·16-s + (1.03 + 0.594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.277 + 0.960i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ -0.277 + 0.960i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.582921 - 0.774899i\)
\(L(\frac12)\) \(\approx\) \(0.582921 - 0.774899i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.25 + 1.97i)T \)
7 \( 1 + (-1.23 + 6.88i)T \)
good2 \( 1 + 0.513iT - 4T^{2} \)
5 \( 1 + (6.08 + 3.51i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-3.15 + 1.82i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (3.79 + 6.56i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-17.5 - 10.1i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-13.6 - 23.7i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-3.42 - 1.97i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (23.7 + 13.6i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 4.85T + 961T^{2} \)
37 \( 1 + (18.7 + 32.4i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-61.1 + 35.2i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (9.41 - 16.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 23.9iT - 2.20e3T^{2} \)
53 \( 1 + (-23.1 - 13.3i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 - 52.8iT - 3.48e3T^{2} \)
61 \( 1 + 106.T + 3.72e3T^{2} \)
67 \( 1 - 102.T + 4.48e3T^{2} \)
71 \( 1 - 138. iT - 5.04e3T^{2} \)
73 \( 1 + (34.7 - 60.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 23.2T + 6.24e3T^{2} \)
83 \( 1 + (25.9 + 14.9i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-135. + 78.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-2.93 + 5.07i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−14.38325269697503625852137423852, −12.76922024180126195561830225489, −12.11776313793675459379406471915, −11.24881099276213474565613988536, −10.25174446219267997697713202033, −7.87072575415999957983883325061, −7.41274258466812957685862299229, −5.73661583747264119991727001480, −3.86025513523402039957113322984, −1.08356770092452054656093718082, 3.18118459213150800443914694071, 5.09688150166375551874556813437, 6.57614481307445689678508341184, 7.61910355865884830188917194826, 9.361709123769738505915741265613, 10.89767598628718036900631125092, 11.66631022843912218683924029320, 12.13581119498476321502143533852, 14.58969898244636939671587026426, 15.25804263263977551417283190885

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