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Properties

Label 2-28-28.27-c3-0-2
Degree $2$
Conductor $28$
Sign $0.565 - 0.824i$
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  + (−2.41 + 1.47i)2-s + 4.79·3-s + (3.65 − 7.11i)4-s + 17.0i·5-s + (−11.5 + 7.07i)6-s + (18.3 + 2.33i)7-s + (1.65 + 22.5i)8-s − 3.97·9-s + (−25.1 − 41.2i)10-s − 41.4i·11-s + (17.5 − 34.1i)12-s − 45.3i·13-s + (−47.7 + 21.4i)14-s + 81.9i·15-s + (−37.2 − 52.0i)16-s + 28.2i·17-s + ⋯
L(s)  = 1  + (−0.853 + 0.521i)2-s + 0.923·3-s + (0.457 − 0.889i)4-s + 1.52i·5-s + (−0.788 + 0.481i)6-s + (0.992 + 0.126i)7-s + (0.0732 + 0.997i)8-s − 0.147·9-s + (−0.795 − 1.30i)10-s − 1.13i·11-s + (0.422 − 0.821i)12-s − 0.967i·13-s + (−0.912 + 0.409i)14-s + 1.41i·15-s + (−0.582 − 0.813i)16-s + 0.403i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.565 - 0.824i$
Analytic conductor: \(1.65205\)
Root analytic conductor: \(1.28532\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :3/2),\ 0.565 - 0.824i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.960593 + 0.505934i\)
\(L(\frac12)\) \(\approx\) \(0.960593 + 0.505934i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.41 - 1.47i)T \)
7 \( 1 + (-18.3 - 2.33i)T \)
good3 \( 1 - 4.79T + 27T^{2} \)
5 \( 1 - 17.0iT - 125T^{2} \)
11 \( 1 + 41.4iT - 1.33e3T^{2} \)
13 \( 1 + 45.3iT - 2.19e3T^{2} \)
17 \( 1 - 28.2iT - 4.91e3T^{2} \)
19 \( 1 + 41.5T + 6.85e3T^{2} \)
23 \( 1 + 93.9iT - 1.21e4T^{2} \)
29 \( 1 - 27.8T + 2.43e4T^{2} \)
31 \( 1 - 81.4T + 2.97e4T^{2} \)
37 \( 1 - 94.8T + 5.06e4T^{2} \)
41 \( 1 - 227. iT - 6.89e4T^{2} \)
43 \( 1 + 171. iT - 7.95e4T^{2} \)
47 \( 1 - 286.T + 1.03e5T^{2} \)
53 \( 1 + 575.T + 1.48e5T^{2} \)
59 \( 1 + 411.T + 2.05e5T^{2} \)
61 \( 1 - 778. iT - 2.26e5T^{2} \)
67 \( 1 + 198. iT - 3.00e5T^{2} \)
71 \( 1 + 197. iT - 3.57e5T^{2} \)
73 \( 1 + 255. iT - 3.89e5T^{2} \)
79 \( 1 - 1.17e3iT - 4.93e5T^{2} \)
83 \( 1 + 938.T + 5.71e5T^{2} \)
89 \( 1 + 1.16e3iT - 7.04e5T^{2} \)
97 \( 1 - 656. iT - 9.12e5T^{2} \)
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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

−17.14104383010778718829746472696, −15.38348077779876654972407563875, −14.65465888818022407623206033554, −13.94147825199111461388058338528, −11.24714635778300143825514149393, −10.43564344324314236145253542904, −8.616673585881805295965825832806, −7.75966662624519035024875089350, −6.08054719674180413494880607669, −2.74031707088156760222472367646, 1.80429656819050611382868374484, 4.46880084781705925586991738794, 7.66656253026416695120763702419, 8.713621286185369342635717612134, 9.542361998127801261337529105601, 11.49399383120051440443400144331, 12.59841873543573690361887382827, 13.96237853127009830601449595016, 15.50932352435908006359974853742, 16.89738113654140841953818268406

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