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Properties

Label 2-648-216.13-c1-0-28
Degree $2$
Conductor $648$
Sign $0.586 + 0.809i$
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  + (1.34 − 0.441i)2-s + (1.60 − 1.18i)4-s + (0.721 − 0.860i)5-s + (1.31 + 0.479i)7-s + (1.63 − 2.30i)8-s + (0.589 − 1.47i)10-s + (−1.05 − 1.26i)11-s + (−0.671 − 0.118i)13-s + (1.98 + 0.0622i)14-s + (1.18 − 3.82i)16-s + (−0.907 + 1.57i)17-s + (2.96 − 1.71i)19-s + (0.140 − 2.24i)20-s + (−1.97 − 1.22i)22-s + (3.62 − 1.31i)23-s + ⋯
L(s)  = 1  + (0.949 − 0.312i)2-s + (0.804 − 0.593i)4-s + (0.322 − 0.384i)5-s + (0.498 + 0.181i)7-s + (0.579 − 0.815i)8-s + (0.186 − 0.466i)10-s + (−0.319 − 0.380i)11-s + (−0.186 − 0.0328i)13-s + (0.529 + 0.0166i)14-s + (0.295 − 0.955i)16-s + (−0.220 + 0.381i)17-s + (0.680 − 0.393i)19-s + (0.0315 − 0.501i)20-s + (−0.421 − 0.261i)22-s + (0.755 − 0.274i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.586 + 0.809i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.586 + 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.58049 - 1.31666i\)
\(L(\frac12)\) \(\approx\) \(2.58049 - 1.31666i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 + 0.441i)T \)
3 \( 1 \)
good5 \( 1 + (-0.721 + 0.860i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (-1.31 - 0.479i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (1.05 + 1.26i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.671 + 0.118i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.907 - 1.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.96 + 1.71i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.62 + 1.31i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (3.92 - 0.691i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (6.21 - 2.26i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-3.83 - 2.21i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.952 - 5.40i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (7.26 + 8.65i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-11.5 - 4.20i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 6.89iT - 53T^{2} \)
59 \( 1 + (-6.78 + 8.09i)T + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.24 + 3.40i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (11.9 + 2.10i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (6.80 - 11.7i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.73 - 8.20i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.59 - 9.03i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.29 - 0.228i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (-3.52 - 6.10i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.33 - 5.31i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.74741160756790377104056702040, −9.663405898995435346135828055160, −8.799110069242941702367229611455, −7.62078142314617763486127356619, −6.71239396212516780510782717623, −5.48620338705365594308934668610, −5.09363376026378072619380455578, −3.83751514431339554423225575872, −2.67121027223647522167114331311, −1.38827624571116212475343943147, 1.94493114602653349759693541804, 3.08352963341420159934392921159, 4.28378192405202149623928219631, 5.21433292154515604937873908425, 6.04623848629487401758904926490, 7.20024302057655413186466805163, 7.63199715584993744265029451610, 8.856964076883014034288935135528, 9.997101547793397133441919762376, 10.88693602217268664808811624043

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