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Properties

Label 2-648-216.157-c1-0-12
Degree $2$
Conductor $648$
Sign $0.822 - 0.568i$
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  + (−0.138 + 1.40i)2-s + (−1.96 − 0.389i)4-s + (−0.608 − 1.67i)5-s + (0.408 + 2.31i)7-s + (0.820 − 2.70i)8-s + (2.43 − 0.624i)10-s + (0.860 − 2.36i)11-s + (0.359 − 0.428i)13-s + (−3.31 + 0.253i)14-s + (3.69 + 1.52i)16-s + (−1.49 + 2.59i)17-s + (6.64 − 3.83i)19-s + (0.541 + 3.51i)20-s + (3.20 + 1.53i)22-s + (1.12 − 6.39i)23-s + ⋯
L(s)  = 1  + (−0.0979 + 0.995i)2-s + (−0.980 − 0.194i)4-s + (−0.272 − 0.747i)5-s + (0.154 + 0.874i)7-s + (0.290 − 0.956i)8-s + (0.770 − 0.197i)10-s + (0.259 − 0.712i)11-s + (0.0997 − 0.118i)13-s + (−0.885 + 0.0678i)14-s + (0.923 + 0.382i)16-s + (−0.363 + 0.629i)17-s + (1.52 − 0.880i)19-s + (0.121 + 0.786i)20-s + (0.683 + 0.327i)22-s + (0.235 − 1.33i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.822 - 0.568i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.822 - 0.568i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20856 + 0.377199i\)
\(L(\frac12)\) \(\approx\) \(1.20856 + 0.377199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.138 - 1.40i)T \)
3 \( 1 \)
good5 \( 1 + (0.608 + 1.67i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.408 - 2.31i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-0.860 + 2.36i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.359 + 0.428i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (1.49 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.64 + 3.83i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.12 + 6.39i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-5.22 - 6.22i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.643 - 3.65i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-4.07 - 2.35i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.67 - 4.76i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-0.928 + 2.55i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.0919 + 0.521i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 6.39iT - 53T^{2} \)
59 \( 1 + (2.87 + 7.88i)T + (-45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.716 + 0.126i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-3.68 + 4.39i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (7.38 - 12.7i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.98 + 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.94 - 7.50i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (4.81 + 5.74i)T + (-14.4 + 81.7i)T^{2} \)
89 \( 1 + (-1.16 - 2.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.8 - 5.05i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.50815580961955893562271342796, −9.354684580930100507017531347495, −8.648327614755971723807254732415, −8.302473299307391452576432413284, −7.04607542533853859571613122742, −6.15802951256226958012786129327, −5.20482215359926891656842612466, −4.52041024028207014204494971136, −3.07093005484320774133020310469, −0.941219951037905073719196057740, 1.18142630384239624558832240366, 2.69305073549330007810321872176, 3.73930167360538921596911235554, 4.55108713633102700257884658129, 5.82718989121485559202282878735, 7.43614394580341594356672671950, 7.54751324975566788215779105402, 9.085130081200160389848980983863, 9.814472231990977725950530162764, 10.47013687692734842106290997352

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