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Properties

Label 2-648-216.13-c1-0-0
Degree $2$
Conductor $648$
Sign $-0.323 - 0.946i$
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  + (−0.495 − 1.32i)2-s + (−1.50 + 1.31i)4-s + (0.185 − 0.221i)5-s + (−2.62 − 0.956i)7-s + (2.48 + 1.35i)8-s + (−0.384 − 0.136i)10-s + (−3.08 − 3.67i)11-s + (3.17 + 0.559i)13-s + (0.0337 + 3.95i)14-s + (0.559 − 3.96i)16-s + (−2.81 + 4.87i)17-s + (0.00428 − 0.00247i)19-s + (0.00986 + 0.577i)20-s + (−3.33 + 5.89i)22-s + (−1.84 + 0.670i)23-s + ⋯
L(s)  = 1  + (−0.350 − 0.936i)2-s + (−0.754 + 0.655i)4-s + (0.0829 − 0.0988i)5-s + (−0.993 − 0.361i)7-s + (0.878 + 0.477i)8-s + (−0.121 − 0.0431i)10-s + (−0.928 − 1.10i)11-s + (0.880 + 0.155i)13-s + (0.00902 + 1.05i)14-s + (0.139 − 0.990i)16-s + (−0.682 + 1.18i)17-s + (0.000984 − 0.000568i)19-s + (0.00220 + 0.129i)20-s + (−0.711 + 1.25i)22-s + (−0.384 + 0.139i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.323 - 0.946i$
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: \((2,\ 648,\ (\ :1/2),\ -0.323 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0217085 + 0.0303767i\)
\(L(\frac12)\) \(\approx\) \(0.0217085 + 0.0303767i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.495 + 1.32i)T \)
3 \( 1 \)
good5 \( 1 + (-0.185 + 0.221i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (2.62 + 0.956i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (3.08 + 3.67i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-3.17 - 0.559i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (2.81 - 4.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.00428 + 0.00247i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.84 - 0.670i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (7.88 - 1.38i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (5.45 - 1.98i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (8.20 + 4.73i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.69 - 9.63i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-3.57 - 4.26i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (3.48 + 1.26i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 0.865iT - 53T^{2} \)
59 \( 1 + (-3.57 + 4.25i)T + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.43 + 6.68i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.224 + 0.0396i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (0.429 - 0.744i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.02 + 3.51i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.797 + 4.52i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-4.00 + 0.705i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (-1.77 - 3.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.04 + 5.07i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.94142771618562365974415734266, −10.07611119555292246468019294597, −9.138537792404049380192509035242, −8.497826065479510930674840434900, −7.53586586727957587821131295881, −6.29951324878766774298843841844, −5.29870945019398232652647733059, −3.78155940552644052732806858362, −3.27962302602939236951780093828, −1.72270932696842009773306413708, 0.02182714999464019809898524067, 2.25397584139284101189046225337, 3.81986416914155998467272390915, 5.04898412639350417697481237981, 5.87572274275978537703032561204, 6.85441409521963986117941336416, 7.46825762939582626830093273217, 8.604090014642264848013115707136, 9.327569709581080485828248412025, 10.07723440491189926527906718583

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