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Properties

Label 2-648-72.59-c1-0-19
Degree $2$
Conductor $648$
Sign $0.637 - 0.770i$
Analytic cond. $5.17430$
Root an. cond. $2.27471$
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  + (−0.841 + 1.13i)2-s + (−0.582 − 1.91i)4-s + (−1.53 + 2.66i)5-s + (3.45 − 1.99i)7-s + (2.66 + 0.949i)8-s + (−1.73 − 3.99i)10-s + (1.64 − 0.949i)11-s + (−0.926 − 0.534i)13-s + (−0.642 + 5.60i)14-s + (−3.32 + 2.22i)16-s − 7.08i·17-s + 3.73·19-s + (5.99 + 1.39i)20-s + (−0.305 + 2.66i)22-s + (0.412 − 0.713i)23-s + ⋯
L(s)  = 1  + (−0.595 + 0.803i)2-s + (−0.291 − 0.956i)4-s + (−0.687 + 1.19i)5-s + (1.30 − 0.754i)7-s + (0.941 + 0.335i)8-s + (−0.547 − 1.26i)10-s + (0.495 − 0.286i)11-s + (−0.256 − 0.148i)13-s + (−0.171 + 1.49i)14-s + (−0.830 + 0.556i)16-s − 1.71i·17-s + 0.856·19-s + (1.34 + 0.311i)20-s + (−0.0652 + 0.568i)22-s + (0.0859 − 0.148i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.637 - 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02607 + 0.482624i\)
\(L(\frac12)\) \(\approx\) \(1.02607 + 0.482624i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 - 1.13i)T \)
3 \( 1 \)
good5 \( 1 + (1.53 - 2.66i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.45 + 1.99i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.64 + 0.949i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.926 + 0.534i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.08iT - 17T^{2} \)
19 \( 1 - 3.73T + 19T^{2} \)
23 \( 1 + (-0.412 + 0.713i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.25 - 3.90i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.06 - 2.92i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.91iT - 37T^{2} \)
41 \( 1 + (-3.28 - 1.89i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.48 - 6.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 + (1.64 + 0.949i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.926 - 0.534i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.59 + 7.96i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 5.92T + 73T^{2} \)
79 \( 1 + (5.30 - 3.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.98 - 5.18i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.6iT - 89T^{2} \)
97 \( 1 + (-5.69 - 9.86i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.64270749726031016203290519224, −9.844250723791500257761974451452, −8.743001509514555326859269732545, −7.78512193415903638449134339197, −7.28176636062285269162061174925, −6.62973932249900839262987736193, −5.21640383589024059430488054365, −4.39570306510303861609435227283, −2.93799786403220938999112603562, −1.04972350359854825501723319371, 1.12164330402515723693735825282, 2.19475110017074117986576450091, 3.94968894311483080396974833781, 4.57251314652048229745647266485, 5.66127967805334394498430094769, 7.37181498417561504329560579100, 8.246913837907073247839095152413, 8.586141306447731136057021276919, 9.443706351286352663684866207170, 10.48338235944193976916775417585

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