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Properties

Label 2-648-216.11-c1-0-0
Degree $2$
Conductor $648$
Sign $0.339 - 0.940i$
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  + (0.521 − 1.31i)2-s + (−1.45 − 1.37i)4-s + (−3.07 − 1.11i)5-s + (−1.33 − 0.235i)7-s + (−2.56 + 1.19i)8-s + (−3.07 + 3.45i)10-s + (0.982 + 2.69i)11-s + (2.36 + 2.82i)13-s + (−1.00 + 1.63i)14-s + (0.234 + 3.99i)16-s + (−1.94 + 1.12i)17-s + (−1.22 + 2.12i)19-s + (2.93 + 5.84i)20-s + (4.05 + 0.117i)22-s + (−1.08 − 6.17i)23-s + ⋯
L(s)  = 1  + (0.369 − 0.929i)2-s + (−0.727 − 0.686i)4-s + (−1.37 − 0.500i)5-s + (−0.505 − 0.0891i)7-s + (−0.906 + 0.423i)8-s + (−0.972 + 1.09i)10-s + (0.296 + 0.813i)11-s + (0.656 + 0.782i)13-s + (−0.269 + 0.437i)14-s + (0.0587 + 0.998i)16-s + (−0.470 + 0.271i)17-s + (−0.280 + 0.486i)19-s + (0.657 + 1.30i)20-s + (0.865 + 0.0250i)22-s + (−0.227 − 1.28i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.339 - 0.940i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.339 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.231599 + 0.162654i\)
\(L(\frac12)\) \(\approx\) \(0.231599 + 0.162654i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.521 + 1.31i)T \)
3 \( 1 \)
good5 \( 1 + (3.07 + 1.11i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (1.33 + 0.235i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-0.982 - 2.69i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-2.36 - 2.82i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.94 - 1.12i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.22 - 2.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.08 + 6.17i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-5.00 - 4.19i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (4.65 - 0.820i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (3.70 - 2.14i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.28 + 8.67i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (3.24 - 1.17i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (2.21 - 12.5i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 4.14T + 53T^{2} \)
59 \( 1 + (3.92 - 10.7i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (9.59 + 1.69i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (6.64 - 5.57i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (3.19 + 5.53i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.87 - 4.97i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.31 - 6.33i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-1.56 + 1.85i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (1.06 + 0.612i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-16.9 + 6.17i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.82602972449593700058294974925, −10.08946689178961914525488153547, −8.888232139660403803732374675425, −8.517038275407992818356656959334, −7.15053065958713068494053123623, −6.19338369511997211036787643263, −4.67566548245857408577982820419, −4.19148720177333649269618859276, −3.23644467358433011238616232809, −1.61402307803824407868008929557, 0.13859290225409104156924209394, 3.23925110018907920153478899877, 3.64659380502925223041479496036, 4.88830249876261527783329310489, 6.06421638947127039129760784686, 6.80142942979790810900044387209, 7.72545762214067159269283920136, 8.364299876328747230636117275117, 9.204186835013648742717855427515, 10.44216905302474957000404235746

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