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Properties

Label 2-28-28.3-c3-0-5
Degree $2$
Conductor $28$
Sign $0.914 - 0.404i$
Analytic cond. $1.65205$
Root an. cond. $1.28532$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.19 + 1.78i)2-s + (2.11 − 3.65i)3-s + (1.63 + 7.83i)4-s + (1.03 − 0.596i)5-s + (11.1 − 4.25i)6-s + (−16.7 − 7.86i)7-s + (−10.4 + 20.0i)8-s + (4.58 + 7.93i)9-s + (3.33 + 0.535i)10-s + (−29.6 − 17.0i)11-s + (32.0 + 10.5i)12-s − 56.1i·13-s + (−22.7 − 47.1i)14-s − 5.04i·15-s + (−58.6 + 25.5i)16-s + (19.4 + 11.2i)17-s + ⋯
L(s)  = 1  + (0.775 + 0.630i)2-s + (0.406 − 0.703i)3-s + (0.203 + 0.979i)4-s + (0.0924 − 0.0533i)5-s + (0.759 − 0.289i)6-s + (−0.905 − 0.424i)7-s + (−0.459 + 0.888i)8-s + (0.169 + 0.293i)9-s + (0.105 + 0.0169i)10-s + (−0.811 − 0.468i)11-s + (0.771 + 0.254i)12-s − 1.19i·13-s + (−0.434 − 0.900i)14-s − 0.0867i·15-s + (−0.916 + 0.398i)16-s + (0.277 + 0.160i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.914 - 0.404i$
Analytic conductor: \(1.65205\)
Root analytic conductor: \(1.28532\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :3/2),\ 0.914 - 0.404i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.72321 + 0.364288i\)
\(L(\frac12)\) \(\approx\) \(1.72321 + 0.364288i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.19 - 1.78i)T \)
7 \( 1 + (16.7 + 7.86i)T \)
good3 \( 1 + (-2.11 + 3.65i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-1.03 + 0.596i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (29.6 + 17.0i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 56.1iT - 2.19e3T^{2} \)
17 \( 1 + (-19.4 - 11.2i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-53.4 - 92.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-51.2 + 29.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 211.T + 2.43e4T^{2} \)
31 \( 1 + (54.1 - 93.8i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (81.0 + 140. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 414. iT - 6.89e4T^{2} \)
43 \( 1 - 258. iT - 7.95e4T^{2} \)
47 \( 1 + (304. + 527. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (112. - 194. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-42.9 + 74.3i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (312. - 180. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-580. - 334. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 133. iT - 3.57e5T^{2} \)
73 \( 1 + (675. + 389. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (729. - 421. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 432.T + 5.71e5T^{2} \)
89 \( 1 + (439. - 253. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 372. iT - 9.12e5T^{2} \)
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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

−16.50685798713088618050443521314, −15.63090417487360778649584266358, −14.09309494268505056109821628458, −13.20634899133848667349996932352, −12.46386630838486728514193430257, −10.41002352981097223431908703083, −8.241320225459822256534138518992, −7.19288879272450215878265348051, −5.56022221124572662895955781957, −3.18081444227995111708802286805, 2.94544014725477173632933558786, 4.66724084565712087755271352731, 6.58506647697982449272937165328, 9.295746432936106375433274540747, 10.06849924037960167001689918015, 11.71188121323861134808277901679, 12.93555291048904178099802085551, 14.13735245177399291525438424223, 15.41586218549616971705534384361, 16.04108126642001174538862559289

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