Properties

Label 2-63-9.2-c2-0-1
Degree $2$
Conductor $63$
Sign $0.129 - 0.991i$
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  + (−2.50 − 1.44i)2-s + (2.16 + 2.07i)3-s + (2.19 + 3.79i)4-s + (−6.82 + 3.94i)5-s + (−2.43 − 8.33i)6-s + (−1.32 + 2.29i)7-s − 1.10i·8-s + (0.404 + 8.99i)9-s + 22.8·10-s + (1.97 + 1.13i)11-s + (−3.11 + 12.7i)12-s + (7.03 + 12.1i)13-s + (6.63 − 3.83i)14-s + (−22.9 − 5.60i)15-s + (7.16 − 12.4i)16-s − 5.53i·17-s + ⋯
L(s)  = 1  + (−1.25 − 0.723i)2-s + (0.722 + 0.691i)3-s + (0.547 + 0.948i)4-s + (−1.36 + 0.788i)5-s + (−0.406 − 1.38i)6-s + (−0.188 + 0.327i)7-s − 0.138i·8-s + (0.0449 + 0.998i)9-s + 2.28·10-s + (0.179 + 0.103i)11-s + (−0.259 + 1.06i)12-s + (0.540 + 0.936i)13-s + (0.473 − 0.273i)14-s + (−1.53 − 0.373i)15-s + (0.447 − 0.775i)16-s − 0.325i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.129 - 0.991i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ 0.129 - 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.405148 + 0.355780i\)
\(L(\frac12)\) \(\approx\) \(0.405148 + 0.355780i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.16 - 2.07i)T \)
7 \( 1 + (1.32 - 2.29i)T \)
good2 \( 1 + (2.50 + 1.44i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (6.82 - 3.94i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-1.97 - 1.13i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-7.03 - 12.1i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 5.53iT - 289T^{2} \)
19 \( 1 + 33.1T + 361T^{2} \)
23 \( 1 + (-21.8 + 12.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-9.48 - 5.47i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-13.5 - 23.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 0.870T + 1.36e3T^{2} \)
41 \( 1 + (-3.05 + 1.76i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-31.7 + 54.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-72.7 - 42.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 33.8iT - 2.80e3T^{2} \)
59 \( 1 + (-2.45 + 1.41i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (33.3 - 57.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-26.6 - 46.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 73.1iT - 5.04e3T^{2} \)
73 \( 1 - 97.8T + 5.32e3T^{2} \)
79 \( 1 + (-0.527 + 0.913i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-34.2 - 19.7i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 124. iT - 7.92e3T^{2} \)
97 \( 1 + (0.919 - 1.59i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−15.13641938904996401794706470023, −14.17842085084145791617362676469, −12.25833393619687087670491946224, −11.07946324451878504792699428752, −10.53404984178500766201475773152, −9.053227729028892705764259099656, −8.418052176227985352219726311724, −7.08690765460161962174888961710, −4.16054244723427066871014599882, −2.67038888135437030398025172657, 0.70670602665824280633207500771, 3.85904287287016751175984533412, 6.48652581470176288063738473151, 7.77337848848608240696100230291, 8.288402652414922054760265870783, 9.216994414997906100210387843717, 10.84046343619125700329578237754, 12.42310760732601448518237847213, 13.20862223589377594912366058887, 15.05505692311804625082112105345

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