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Properties

Label 2-648-216.133-c1-0-4
Degree $2$
Conductor $648$
Sign $0.885 - 0.465i$
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  + (−1.15 − 0.820i)2-s + (0.652 + 1.89i)4-s + (−2.15 − 2.57i)5-s + (−1.78 + 0.649i)7-s + (0.799 − 2.71i)8-s + (0.374 + 4.73i)10-s + (−3.21 + 3.83i)11-s + (3.61 − 0.637i)13-s + (2.58 + 0.716i)14-s + (−3.14 + 2.46i)16-s + (1.74 + 3.02i)17-s + (−2.73 − 1.57i)19-s + (3.45 − 5.76i)20-s + (6.85 − 1.77i)22-s + (2.93 + 1.06i)23-s + ⋯
L(s)  = 1  + (−0.814 − 0.580i)2-s + (0.326 + 0.945i)4-s + (−0.965 − 1.15i)5-s + (−0.674 + 0.245i)7-s + (0.282 − 0.959i)8-s + (0.118 + 1.49i)10-s + (−0.970 + 1.15i)11-s + (1.00 − 0.176i)13-s + (0.691 + 0.191i)14-s + (−0.786 + 0.617i)16-s + (0.422 + 0.732i)17-s + (−0.626 − 0.361i)19-s + (0.772 − 1.28i)20-s + (1.46 − 0.378i)22-s + (0.611 + 0.222i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.885 - 0.465i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.885 - 0.465i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.540989 + 0.133473i\)
\(L(\frac12)\) \(\approx\) \(0.540989 + 0.133473i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 + 0.820i)T \)
3 \( 1 \)
good5 \( 1 + (2.15 + 2.57i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (1.78 - 0.649i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (3.21 - 3.83i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-3.61 + 0.637i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-1.74 - 3.02i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.73 + 1.57i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.93 - 1.06i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-6.61 - 1.16i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-0.842 - 0.306i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-6.21 + 3.58i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.341 - 1.93i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (5.36 - 6.39i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (2.63 - 0.959i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 - 9.52iT - 53T^{2} \)
59 \( 1 + (-6.13 - 7.31i)T + (-10.2 + 58.1i)T^{2} \)
61 \( 1 + (-2.92 - 8.03i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (-13.7 + 2.42i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (1.50 + 2.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.472 + 0.818i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.803 - 4.55i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (2.12 + 0.375i)T + (77.9 + 28.3i)T^{2} \)
89 \( 1 + (-7.83 + 13.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.68 - 1.41i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.54376668358847407375764433309, −9.743119039187481634872722590877, −8.818860681185247783310996575487, −8.220331002115608985703849353421, −7.49011896086808298418981665809, −6.32067458180897972863371856124, −4.83428790358786875568611338016, −3.94805523798003974704260202237, −2.74579253094685749991623727674, −1.12522079389807038221413626653, 0.47115360144674441090256775920, 2.74334920218141633996696272598, 3.65614627756699488639856983432, 5.26593778036797746194138739939, 6.48358145811990741147600995358, 6.82491714778207266784735644430, 8.096930372933873586135013926401, 8.335401663262138933135439663187, 9.713827079298190042205466483196, 10.49827626562350420903863494368

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