Properties

Degree $2$
Conductor $63$
Sign $0.787 + 0.616i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.119 + 0.207i)2-s + (−0.619 − 1.61i)3-s + (0.971 − 1.68i)4-s + (−0.590 + 1.02i)5-s + (0.260 − 0.321i)6-s + (0.5 + 0.866i)7-s + 0.942·8-s + (−2.23 + 2.00i)9-s − 0.282·10-s + (1.85 + 3.20i)11-s + (−3.32 − 0.528i)12-s + (−0.5 + 0.866i)13-s + (−0.119 + 0.207i)14-s + (2.02 + 0.321i)15-s + (−1.83 − 3.16i)16-s − 6.94·17-s + ⋯
L(s)  = 1  + (0.0845 + 0.146i)2-s + (−0.357 − 0.933i)3-s + (0.485 − 0.841i)4-s + (−0.264 + 0.457i)5-s + (0.106 − 0.131i)6-s + (0.188 + 0.327i)7-s + 0.333·8-s + (−0.744 + 0.668i)9-s − 0.0893·10-s + (0.558 + 0.967i)11-s + (−0.959 − 0.152i)12-s + (−0.138 + 0.240i)13-s + (−0.0319 + 0.0553i)14-s + (0.522 + 0.0830i)15-s + (−0.457 − 0.792i)16-s − 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.787 + 0.616i$
Motivic weight: \(1\)
Character: $\chi_{63} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.787 + 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.839017 - 0.289564i\)
\(L(\frac12)\) \(\approx\) \(0.839017 - 0.289564i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.619 + 1.61i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.119 - 0.207i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.590 - 1.02i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.85 - 3.20i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.94T + 17T^{2} \)
19 \( 1 - 1.94T + 19T^{2} \)
23 \( 1 + (-2.80 + 4.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.119 + 0.207i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.830 - 1.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.54T + 37T^{2} \)
41 \( 1 + (-5.09 + 8.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.11 + 1.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.91 + 5.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + (1.30 - 2.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.80 - 6.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.75 - 3.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.60T + 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 + (3.68 + 6.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.47 - 6.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.74T + 89T^{2} \)
97 \( 1 + (3.58 + 6.20i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.82703427831265905370011923616, −13.91397579454993627414007010512, −12.54563433457832995745980727043, −11.47976323945043829105803933202, −10.64813676893796447985492551445, −8.982112909699026641535673828233, −7.16857274627485590499394876326, −6.58896046172724283742157308827, −5.01523058526717656416219994446, −2.09213707624128707708651459098, 3.40962639127877897477938134139, 4.70683705393440573712557272699, 6.52446111003679327033583646723, 8.192470646892910787585094736683, 9.263132994305253607061786782226, 10.98262122079192453534303582760, 11.44950313787024898847570477222, 12.72063900394148063617183274025, 13.97777342723605021534233627437, 15.47443306469881623533155391655

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