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Properties

Label 2-648-648.155-c1-0-41
Degree $2$
Conductor $648$
Sign $-0.180 - 0.983i$
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 + 0.196i)2-s + (−0.739 + 1.56i)3-s + (1.92 + 0.551i)4-s + (−0.405 + 0.266i)5-s + (−1.34 + 2.04i)6-s + (0.678 + 0.639i)7-s + (2.58 + 1.15i)8-s + (−1.90 − 2.31i)9-s + (−0.621 + 0.293i)10-s + (1.65 + 3.30i)11-s + (−2.28 + 2.60i)12-s + (−1.97 + 0.852i)13-s + (0.823 + 1.02i)14-s + (−0.117 − 0.833i)15-s + (3.39 + 2.12i)16-s + (−0.154 − 0.184i)17-s + ⋯
L(s)  = 1  + (0.990 + 0.139i)2-s + (−0.427 + 0.904i)3-s + (0.961 + 0.275i)4-s + (−0.181 + 0.119i)5-s + (−0.548 + 0.835i)6-s + (0.256 + 0.241i)7-s + (0.913 + 0.406i)8-s + (−0.635 − 0.772i)9-s + (−0.196 + 0.0929i)10-s + (0.499 + 0.995i)11-s + (−0.659 + 0.751i)12-s + (−0.548 + 0.236i)13-s + (0.220 + 0.275i)14-s + (−0.0304 − 0.215i)15-s + (0.847 + 0.530i)16-s + (−0.0374 − 0.0446i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.180 - 0.983i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ -0.180 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49974 + 1.80038i\)
\(L(\frac12)\) \(\approx\) \(1.49974 + 1.80038i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.196i)T \)
3 \( 1 + (0.739 - 1.56i)T \)
good5 \( 1 + (0.405 - 0.266i)T + (1.98 - 4.59i)T^{2} \)
7 \( 1 + (-0.678 - 0.639i)T + (0.407 + 6.98i)T^{2} \)
11 \( 1 + (-1.65 - 3.30i)T + (-6.56 + 8.82i)T^{2} \)
13 \( 1 + (1.97 - 0.852i)T + (8.92 - 9.45i)T^{2} \)
17 \( 1 + (0.154 + 0.184i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (-2.97 - 2.49i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (0.582 + 0.617i)T + (-1.33 + 22.9i)T^{2} \)
29 \( 1 + (7.65 - 0.894i)T + (28.2 - 6.68i)T^{2} \)
31 \( 1 + (2.31 + 9.77i)T + (-27.7 + 13.9i)T^{2} \)
37 \( 1 + (-6.33 + 1.11i)T + (34.7 - 12.6i)T^{2} \)
41 \( 1 + (-6.78 - 5.05i)T + (11.7 + 39.2i)T^{2} \)
43 \( 1 + (-0.111 + 1.90i)T + (-42.7 - 4.99i)T^{2} \)
47 \( 1 + (6.24 + 1.48i)T + (42.0 + 21.0i)T^{2} \)
53 \( 1 + (-0.422 - 0.731i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.02 + 2.03i)T + (-35.2 - 47.3i)T^{2} \)
61 \( 1 + (-3.82 + 1.14i)T + (50.9 - 33.5i)T^{2} \)
67 \( 1 + (-13.3 - 1.56i)T + (65.1 + 15.4i)T^{2} \)
71 \( 1 + (-4.03 - 1.46i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (1.45 - 0.530i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-12.3 + 9.21i)T + (22.6 - 75.6i)T^{2} \)
83 \( 1 + (-3.38 + 2.52i)T + (23.8 - 79.5i)T^{2} \)
89 \( 1 + (3.16 + 8.69i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (13.0 + 8.58i)T + (38.4 + 89.0i)T^{2} \)
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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

−11.27267370576464302808364067669, −9.886754271161662803537436747875, −9.424945162535065326337115831963, −7.921778628618135146302711923432, −7.11539540395141906683924974579, −6.00249250899182821366854123879, −5.24095593069232076677559765636, −4.31072663604345694445766601829, −3.57227863086426991917484729059, −2.10586165365187983389376004818, 1.02971007299898553317648183382, 2.45266719185129792722703571230, 3.66663895553842186557669852390, 4.93379953259972411973461560829, 5.72943997647921645933657806552, 6.61430182146315716930918562536, 7.45162378530700065726138694445, 8.226437084987120794955253116610, 9.564574406327489741153498328197, 10.92660009542824505241664309672

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