Properties

Label 2-648-648.59-c1-0-38
Degree $2$
Conductor $648$
Sign $-0.912 - 0.408i$
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  + (0.949 + 1.04i)2-s + (1.35 + 1.08i)3-s + (−0.197 + 1.99i)4-s + (−2.04 + 0.238i)5-s + (0.145 + 2.44i)6-s + (−0.409 + 0.815i)7-s + (−2.27 + 1.68i)8-s + (0.648 + 2.92i)9-s + (−2.18 − 1.91i)10-s + (2.73 − 1.18i)11-s + (−2.42 + 2.47i)12-s + (−0.613 + 2.59i)13-s + (−1.24 + 0.344i)14-s + (−3.01 − 1.89i)15-s + (−3.92 − 0.787i)16-s + (1.01 + 0.178i)17-s + ⋯
L(s)  = 1  + (0.671 + 0.741i)2-s + (0.779 + 0.625i)3-s + (−0.0988 + 0.995i)4-s + (−0.912 + 0.106i)5-s + (0.0594 + 0.998i)6-s + (−0.154 + 0.308i)7-s + (−0.803 + 0.594i)8-s + (0.216 + 0.976i)9-s + (−0.691 − 0.604i)10-s + (0.826 − 0.356i)11-s + (−0.700 + 0.714i)12-s + (−0.170 + 0.718i)13-s + (−0.332 + 0.0921i)14-s + (−0.778 − 0.488i)15-s + (−0.980 − 0.196i)16-s + (0.245 + 0.0432i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.912 - 0.408i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ -0.912 - 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.445965 + 2.09089i\)
\(L(\frac12)\) \(\approx\) \(0.445965 + 2.09089i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.949 - 1.04i)T \)
3 \( 1 + (-1.35 - 1.08i)T \)
good5 \( 1 + (2.04 - 0.238i)T + (4.86 - 1.15i)T^{2} \)
7 \( 1 + (0.409 - 0.815i)T + (-4.18 - 5.61i)T^{2} \)
11 \( 1 + (-2.73 + 1.18i)T + (7.54 - 8.00i)T^{2} \)
13 \( 1 + (0.613 - 2.59i)T + (-11.6 - 5.83i)T^{2} \)
17 \( 1 + (-1.01 - 0.178i)T + (15.9 + 5.81i)T^{2} \)
19 \( 1 + (0.421 + 2.39i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (-0.183 + 0.0922i)T + (13.7 - 18.4i)T^{2} \)
29 \( 1 + (-1.31 + 4.40i)T + (-24.2 - 15.9i)T^{2} \)
31 \( 1 + (2.30 - 3.50i)T + (-12.2 - 28.4i)T^{2} \)
37 \( 1 + (-0.675 - 1.85i)T + (-28.3 + 23.7i)T^{2} \)
41 \( 1 + (-3.98 - 3.76i)T + (2.38 + 40.9i)T^{2} \)
43 \( 1 + (0.203 - 0.272i)T + (-12.3 - 41.1i)T^{2} \)
47 \( 1 + (-2.49 + 1.64i)T + (18.6 - 43.1i)T^{2} \)
53 \( 1 + (-3.47 + 6.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-13.9 - 6.00i)T + (40.4 + 42.9i)T^{2} \)
61 \( 1 + (-10.6 + 0.618i)T + (60.5 - 7.08i)T^{2} \)
67 \( 1 + (1.96 + 6.55i)T + (-55.9 + 36.8i)T^{2} \)
71 \( 1 + (-10.3 - 8.71i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-4.32 + 3.62i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (4.66 - 4.40i)T + (4.59 - 78.8i)T^{2} \)
83 \( 1 + (-5.09 + 4.80i)T + (4.82 - 82.8i)T^{2} \)
89 \( 1 + (1.51 + 1.80i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (15.1 + 1.76i)T + (94.3 + 22.3i)T^{2} \)
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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

−11.22042403464457126266392716135, −9.796258210681983688651630198887, −8.930381771225153919146407037290, −8.312088278269492220615820055975, −7.41468206778274170907332025554, −6.57913483665797203226549507084, −5.33512003893734010142552437143, −4.22788534302613961622846997814, −3.71277865704528799582747697694, −2.55626907930297348751487056957, 0.889393134300606800148372479037, 2.32204536594915203670085712905, 3.61938278898258296359102544921, 4.04806562457889839499470876227, 5.50818792012468090503097091261, 6.69298908551241252355446989874, 7.49571474168505860919399879699, 8.466204247505718882583669377734, 9.408843373763013738025755637529, 10.21708205945885139365102990922

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