Properties

Label 2-648-648.155-c1-0-31
Degree $2$
Conductor $648$
Sign $0.995 + 0.0915i$
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  + (−1.10 + 0.884i)2-s + (−1.15 + 1.29i)3-s + (0.435 − 1.95i)4-s + (−3.47 + 2.28i)5-s + (0.129 − 2.44i)6-s + (−0.348 − 0.328i)7-s + (1.24 + 2.53i)8-s + (−0.340 − 2.98i)9-s + (1.81 − 5.59i)10-s + (−0.145 − 0.289i)11-s + (2.02 + 2.81i)12-s + (−3.51 + 1.51i)13-s + (0.674 + 0.0545i)14-s + (1.05 − 7.12i)15-s + (−3.62 − 1.69i)16-s + (−0.326 − 0.388i)17-s + ⋯
L(s)  = 1  + (−0.780 + 0.625i)2-s + (−0.665 + 0.746i)3-s + (0.217 − 0.976i)4-s + (−1.55 + 1.02i)5-s + (0.0527 − 0.998i)6-s + (−0.131 − 0.124i)7-s + (0.440 + 0.897i)8-s + (−0.113 − 0.993i)9-s + (0.572 − 1.76i)10-s + (−0.0437 − 0.0872i)11-s + (0.583 + 0.812i)12-s + (−0.975 + 0.420i)13-s + (0.180 + 0.0145i)14-s + (0.271 − 1.83i)15-s + (−0.905 − 0.424i)16-s + (−0.0791 − 0.0942i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.995 + 0.0915i$
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: \((2,\ 648,\ (\ :1/2),\ 0.995 + 0.0915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.224328 - 0.0102893i\)
\(L(\frac12)\) \(\approx\) \(0.224328 - 0.0102893i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.10 - 0.884i)T \)
3 \( 1 + (1.15 - 1.29i)T \)
good5 \( 1 + (3.47 - 2.28i)T + (1.98 - 4.59i)T^{2} \)
7 \( 1 + (0.348 + 0.328i)T + (0.407 + 6.98i)T^{2} \)
11 \( 1 + (0.145 + 0.289i)T + (-6.56 + 8.82i)T^{2} \)
13 \( 1 + (3.51 - 1.51i)T + (8.92 - 9.45i)T^{2} \)
17 \( 1 + (0.326 + 0.388i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (5.23 + 4.38i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (-3.08 - 3.26i)T + (-1.33 + 22.9i)T^{2} \)
29 \( 1 + (1.89 - 0.221i)T + (28.2 - 6.68i)T^{2} \)
31 \( 1 + (-1.59 - 6.72i)T + (-27.7 + 13.9i)T^{2} \)
37 \( 1 + (-9.87 + 1.74i)T + (34.7 - 12.6i)T^{2} \)
41 \( 1 + (-5.36 - 3.99i)T + (11.7 + 39.2i)T^{2} \)
43 \( 1 + (0.0963 - 1.65i)T + (-42.7 - 4.99i)T^{2} \)
47 \( 1 + (4.08 + 0.969i)T + (42.0 + 21.0i)T^{2} \)
53 \( 1 + (4.33 + 7.51i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.18 - 6.34i)T + (-35.2 - 47.3i)T^{2} \)
61 \( 1 + (-3.63 + 1.08i)T + (50.9 - 33.5i)T^{2} \)
67 \( 1 + (-11.9 - 1.40i)T + (65.1 + 15.4i)T^{2} \)
71 \( 1 + (8.64 + 3.14i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-4.71 + 1.71i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (3.75 - 2.79i)T + (22.6 - 75.6i)T^{2} \)
83 \( 1 + (-8.99 + 6.69i)T + (23.8 - 79.5i)T^{2} \)
89 \( 1 + (1.50 + 4.14i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (0.226 + 0.148i)T + (38.4 + 89.0i)T^{2} \)
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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

−10.57886991583747132891776077483, −9.733957579197797618149728331452, −8.820832786701486892279555325921, −7.79912203356391772309674709985, −6.94891274095200375328510989848, −6.45974530819067343772145788501, −4.99873813493536039934942684815, −4.20150975897966853711666775083, −2.85996649242310345647808692852, −0.24101237046949560311838053812, 0.868898499843223610243323276496, 2.44937587822467310013970318562, 4.01664102792310864619462984782, 4.82422059260986833917461376284, 6.29031798079359155344889303592, 7.54320415549825775440392045794, 7.87316270311071603636009612559, 8.655123592712373355231067711493, 9.715793478278538077220007621640, 10.87158878769850394373036334859

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