Properties

Label 2-648-648.133-c1-0-40
Degree $2$
Conductor $648$
Sign $-0.0406 - 0.999i$
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  + (1.26 + 0.624i)2-s + (−1.71 + 0.266i)3-s + (1.21 + 1.58i)4-s + (−0.413 + 0.390i)5-s + (−2.33 − 0.730i)6-s + (2.58 − 0.302i)7-s + (0.555 + 2.77i)8-s + (2.85 − 0.913i)9-s + (−0.768 + 0.236i)10-s + (−2.39 + 0.716i)11-s + (−2.50 − 2.38i)12-s + (4.85 − 0.283i)13-s + (3.47 + 1.23i)14-s + (0.603 − 0.778i)15-s + (−1.02 + 3.86i)16-s + (−2.47 − 0.901i)17-s + ⋯
L(s)  = 1  + (0.897 + 0.441i)2-s + (−0.988 + 0.154i)3-s + (0.609 + 0.792i)4-s + (−0.184 + 0.174i)5-s + (−0.954 − 0.298i)6-s + (0.978 − 0.114i)7-s + (0.196 + 0.980i)8-s + (0.952 − 0.304i)9-s + (−0.243 + 0.0748i)10-s + (−0.721 + 0.216i)11-s + (−0.724 − 0.689i)12-s + (1.34 − 0.0784i)13-s + (0.928 + 0.329i)14-s + (0.155 − 0.200i)15-s + (−0.256 + 0.966i)16-s + (−0.600 − 0.218i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0406 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0406 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.0406 - 0.999i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ -0.0406 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36578 + 1.42253i\)
\(L(\frac12)\) \(\approx\) \(1.36578 + 1.42253i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 - 0.624i)T \)
3 \( 1 + (1.71 - 0.266i)T \)
good5 \( 1 + (0.413 - 0.390i)T + (0.290 - 4.99i)T^{2} \)
7 \( 1 + (-2.58 + 0.302i)T + (6.81 - 1.61i)T^{2} \)
11 \( 1 + (2.39 - 0.716i)T + (9.19 - 6.04i)T^{2} \)
13 \( 1 + (-4.85 + 0.283i)T + (12.9 - 1.50i)T^{2} \)
17 \( 1 + (2.47 + 0.901i)T + (13.0 + 10.9i)T^{2} \)
19 \( 1 + (-2.24 - 6.16i)T + (-14.5 + 12.2i)T^{2} \)
23 \( 1 + (2.42 + 0.283i)T + (22.3 + 5.30i)T^{2} \)
29 \( 1 + (0.244 + 0.487i)T + (-17.3 + 23.2i)T^{2} \)
31 \( 1 + (2.69 - 3.62i)T + (-8.89 - 29.6i)T^{2} \)
37 \( 1 + (0.893 + 1.06i)T + (-6.42 + 36.4i)T^{2} \)
41 \( 1 + (-6.72 + 4.42i)T + (16.2 - 37.6i)T^{2} \)
43 \( 1 + (-2.03 + 8.57i)T + (-38.4 - 19.2i)T^{2} \)
47 \( 1 + (-4.70 - 6.31i)T + (-13.4 + 45.0i)T^{2} \)
53 \( 1 + (-3.99 + 2.30i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.622 + 0.186i)T + (49.2 + 32.4i)T^{2} \)
61 \( 1 + (-7.38 - 3.18i)T + (41.8 + 44.3i)T^{2} \)
67 \( 1 + (2.13 - 4.24i)T + (-40.0 - 53.7i)T^{2} \)
71 \( 1 + (1.66 + 9.46i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-1.34 + 7.63i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (11.3 + 7.45i)T + (31.2 + 72.5i)T^{2} \)
83 \( 1 + (-1.29 + 1.97i)T + (-32.8 - 76.2i)T^{2} \)
89 \( 1 + (-0.682 + 3.87i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-11.3 + 12.0i)T + (-5.64 - 96.8i)T^{2} \)
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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

−10.95461714593233247188011480906, −10.40605509211599000906692629364, −8.842932148141971953418582230138, −7.78391087343233312237717357697, −7.18252817738188373898166410672, −5.95290882095426895858084478339, −5.45749095566742761384344202076, −4.41253961822209110398269109955, −3.57839592119324452674890375030, −1.74250175931274683308094376047, 0.993289237502235467326570010591, 2.36726846029656513270116373749, 4.02491673964744985470131649596, 4.79171038407164220422325307648, 5.63657612754066671749104198537, 6.42410195586334740345596155570, 7.48253699613934382913651963721, 8.526026826557055000214809772241, 9.825235966033722911165314797324, 10.93818700324289668079526862119

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