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Properties

Label 2-63-63.58-c1-0-5
Degree $2$
Conductor $63$
Sign $0.888 + 0.458i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.73i·3-s − 4-s + (0.5 + 0.866i)5-s − 1.73i·6-s + (2 + 1.73i)7-s − 3·8-s − 2.99·9-s + (0.5 + 0.866i)10-s + (−2.5 + 4.33i)11-s + 1.73i·12-s + (2.5 − 4.33i)13-s + (2 + 1.73i)14-s + (1.49 − 0.866i)15-s − 16-s + (−1.5 − 2.59i)17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.999i·3-s − 0.5·4-s + (0.223 + 0.387i)5-s − 0.707i·6-s + (0.755 + 0.654i)7-s − 1.06·8-s − 0.999·9-s + (0.158 + 0.273i)10-s + (−0.753 + 1.30i)11-s + 0.499i·12-s + (0.693 − 1.20i)13-s + (0.534 + 0.462i)14-s + (0.387 − 0.223i)15-s − 0.250·16-s + (−0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.888 + 0.458i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.888 + 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05758 - 0.256465i\)
\(L(\frac12)\) \(\approx\) \(1.05758 - 0.256465i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
7 \( 1 + (-2 - 1.73i)T \)
good2 \( 1 - T + 2T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (1.5 + 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.5 + 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.5 - 7.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.66473203835879405522644063139, −13.71854804953023013846994232051, −12.74061879927651581759917057161, −12.05106951289808834686854428832, −10.52931791595191573691488646241, −8.821719695832177674354867956115, −7.70328543557386928749158187665, −6.08576838044891827252090132427, −4.92887929223857339931243733265, −2.62442707553975429084638472217, 3.67265667809824048879216192133, 4.77534757262665122469953113926, 5.92206473894595284595641012818, 8.349220728424878624324179693032, 9.183117331261402809403179304908, 10.68987656203190730086364755354, 11.57286163977191096296197937749, 13.31543831428165683678846709223, 13.88680957493127461701048482296, 14.86320811388511255413287212482

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