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Properties

Label 2-28-4.3-c4-0-1
Degree $2$
Conductor $28$
Sign $-0.999 + 0.0379i$
Analytic cond. $2.89435$
Root an. cond. $1.70128$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.77 + 2.88i)2-s + 16.9i·3-s + (−0.607 − 15.9i)4-s − 4.68·5-s + (−48.8 − 46.9i)6-s − 18.5i·7-s + (47.7 + 42.6i)8-s − 205.·9-s + (12.9 − 13.4i)10-s + 116. i·11-s + (270. − 10.2i)12-s + 74.0·13-s + (53.3 + 51.3i)14-s − 79.3i·15-s + (−255. + 19.4i)16-s + 152.·17-s + ⋯
L(s)  = 1  + (−0.693 + 0.720i)2-s + 1.88i·3-s + (−0.0379 − 0.999i)4-s − 0.187·5-s + (−1.35 − 1.30i)6-s − 0.377i·7-s + (0.746 + 0.665i)8-s − 2.54·9-s + (0.129 − 0.134i)10-s + 0.960i·11-s + (1.88 − 0.0714i)12-s + 0.438·13-s + (0.272 + 0.262i)14-s − 0.352i·15-s + (−0.997 + 0.0758i)16-s + 0.526·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0379i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $-0.999 + 0.0379i$
Analytic conductor: \(2.89435\)
Root analytic conductor: \(1.70128\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :2),\ -0.999 + 0.0379i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0149523 - 0.787444i\)
\(L(\frac12)\) \(\approx\) \(0.0149523 - 0.787444i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.77 - 2.88i)T \)
7 \( 1 + 18.5iT \)
good3 \( 1 - 16.9iT - 81T^{2} \)
5 \( 1 + 4.68T + 625T^{2} \)
11 \( 1 - 116. iT - 1.46e4T^{2} \)
13 \( 1 - 74.0T + 2.85e4T^{2} \)
17 \( 1 - 152.T + 8.35e4T^{2} \)
19 \( 1 - 329. iT - 1.30e5T^{2} \)
23 \( 1 - 831. iT - 2.79e5T^{2} \)
29 \( 1 - 795.T + 7.07e5T^{2} \)
31 \( 1 + 682. iT - 9.23e5T^{2} \)
37 \( 1 - 1.39e3T + 1.87e6T^{2} \)
41 \( 1 - 2.28e3T + 2.82e6T^{2} \)
43 \( 1 + 286. iT - 3.41e6T^{2} \)
47 \( 1 + 886. iT - 4.87e6T^{2} \)
53 \( 1 - 463.T + 7.89e6T^{2} \)
59 \( 1 + 25.2iT - 1.21e7T^{2} \)
61 \( 1 - 341.T + 1.38e7T^{2} \)
67 \( 1 + 1.96e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.28e3iT - 2.54e7T^{2} \)
73 \( 1 - 3.26e3T + 2.83e7T^{2} \)
79 \( 1 - 1.21e3iT - 3.89e7T^{2} \)
83 \( 1 + 412. iT - 4.74e7T^{2} \)
89 \( 1 + 8.36e3T + 6.27e7T^{2} \)
97 \( 1 - 1.32e4T + 8.85e7T^{2} \)
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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

−16.80535347004331919509715240050, −15.86903061535159500379060312542, −15.11504256823681918076483519772, −14.07025846293443135209482190188, −11.40329856472903767274315826659, −10.14552603276337209203320286420, −9.475797577688729577760772557028, −7.915130545021613079962740841868, −5.67307134812846585817768321440, −4.14489763672864880901890102256, 0.794438058160400155390954967189, 2.68329516209360403330575931660, 6.37908251504027568722918413994, 7.85833410359139689695732826440, 8.781797586501574694715923425403, 11.04931034820709474835675305816, 12.05037986679527079617317055671, 12.99374017279418107064830970845, 14.05897825155567921815112822549, 16.31820798323625333831697387288

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