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Properties

Label 2-648-216.13-c1-0-11
Degree $2$
Conductor $648$
Sign $0.920 + 0.389i$
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  + (−1.40 − 0.118i)2-s + (1.97 + 0.335i)4-s + (−0.508 + 0.606i)5-s + (−3.72 − 1.35i)7-s + (−2.73 − 0.707i)8-s + (0.788 − 0.793i)10-s + (2.92 + 3.49i)11-s + (−3.31 − 0.584i)13-s + (5.08 + 2.35i)14-s + (3.77 + 1.32i)16-s + (2.50 − 4.34i)17-s + (6.10 − 3.52i)19-s + (−1.20 + 1.02i)20-s + (−3.71 − 5.26i)22-s + (4.33 − 1.57i)23-s + ⋯
L(s)  = 1  + (−0.996 − 0.0841i)2-s + (0.985 + 0.167i)4-s + (−0.227 + 0.271i)5-s + (−1.40 − 0.511i)7-s + (−0.968 − 0.250i)8-s + (0.249 − 0.250i)10-s + (0.883 + 1.05i)11-s + (−0.920 − 0.162i)13-s + (1.35 + 0.628i)14-s + (0.943 + 0.330i)16-s + (0.608 − 1.05i)17-s + (1.40 − 0.809i)19-s + (−0.269 + 0.229i)20-s + (−0.791 − 1.12i)22-s + (0.903 − 0.328i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.920 + 0.389i$
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: \((2,\ 648,\ (\ :1/2),\ 0.920 + 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.765736 - 0.155446i\)
\(L(\frac12)\) \(\approx\) \(0.765736 - 0.155446i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.118i)T \)
3 \( 1 \)
good5 \( 1 + (0.508 - 0.606i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (3.72 + 1.35i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-2.92 - 3.49i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (3.31 + 0.584i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-2.50 + 4.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.10 + 3.52i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.33 + 1.57i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (3.32 - 0.585i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-7.51 + 2.73i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-3.56 - 2.05i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.333 - 1.88i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (3.66 + 4.36i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-7.37 - 2.68i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 1.95iT - 53T^{2} \)
59 \( 1 + (-3.02 + 3.60i)T + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.87 + 5.15i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-5.36 - 0.945i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-5.56 + 9.63i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.32 + 2.28i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.150 + 0.852i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.0489 - 0.00863i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (-2.15 - 3.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.00 + 4.19i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.13318323888060600952636092669, −9.583198014552288916555952327496, −9.220101690774209996967811184490, −7.59942559997753305948490543426, −7.13329401375878065614957176965, −6.52801945400909354079267373888, −5.04232647285307807146631351368, −3.49001628710142478163537803012, −2.67142873981136160281560991520, −0.78601652573166509200492927025, 0.995270901082975235256766180393, 2.78563864054033031474662141241, 3.67313126822277226929453413601, 5.56379916158229017774317504712, 6.24690400295123956579127756672, 7.13748094432267818337599437810, 8.174318402150662010758134761000, 8.957975159335609203060144840817, 9.692230081683947439309937136891, 10.25839766569019331100249988306

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