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Properties

Label 2-28-7.4-c3-0-0
Degree $2$
Conductor $28$
Sign $0.245 - 0.969i$
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  + (−3.04 + 5.26i)3-s + (9.58 + 16.5i)5-s + (−6.16 − 17.4i)7-s + (−5 − 8.66i)9-s + (13.2 − 23.0i)11-s + 10.3·13-s − 116.·15-s + (50.6 − 87.7i)17-s + (46.8 + 81.1i)19-s + (110. + 20.6i)21-s + (4.28 + 7.42i)23-s + (−121. + 209. i)25-s − 103.·27-s + 52.3·29-s + (27.7 − 47.9i)31-s + ⋯
L(s)  = 1  + (−0.585 + 1.01i)3-s + (0.857 + 1.48i)5-s + (−0.332 − 0.942i)7-s + (−0.185 − 0.320i)9-s + (0.364 − 0.630i)11-s + 0.220·13-s − 2.00·15-s + (0.722 − 1.25i)17-s + (0.566 + 0.980i)19-s + (1.15 + 0.214i)21-s + (0.0388 + 0.0673i)23-s + (−0.969 + 1.67i)25-s − 0.737·27-s + 0.335·29-s + (0.160 − 0.278i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.245 - 0.969i$
Analytic conductor: \(1.65205\)
Root analytic conductor: \(1.28532\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :3/2),\ 0.245 - 0.969i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.869008 + 0.676677i\)
\(L(\frac12)\) \(\approx\) \(0.869008 + 0.676677i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (6.16 + 17.4i)T \)
good3 \( 1 + (3.04 - 5.26i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-9.58 - 16.5i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-13.2 + 23.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 10.3T + 2.19e3T^{2} \)
17 \( 1 + (-50.6 + 87.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-46.8 - 81.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-4.28 - 7.42i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 52.3T + 2.43e4T^{2} \)
31 \( 1 + (-27.7 + 47.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (214. + 371. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 137.T + 6.89e4T^{2} \)
43 \( 1 + 172T + 7.95e4T^{2} \)
47 \( 1 + (-24.6 - 42.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-237. + 410. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (98.5 - 170. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-200. - 347. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (62.7 - 108. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 788.T + 3.57e5T^{2} \)
73 \( 1 + (302. - 523. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (391. + 678. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 339.T + 5.71e5T^{2} \)
89 \( 1 + (255. + 443. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 672.T + 9.12e5T^{2} \)
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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

−16.84933175403384931276762336929, −15.98283093139049352418133136168, −14.42757227084739258088252109223, −13.70228063829363351229242009006, −11.43864920570591855030025090275, −10.41204732926217839515874382774, −9.759168889496545663055424274214, −7.13620111543091371005115488851, −5.68504951299888844705207226744, −3.50811295432978904342441911189, 1.47758963390656241625822181562, 5.25779715443103296975800903626, 6.46513528920593768258583569211, 8.499630381252146260939173600884, 9.718971425808629794624026418556, 11.99714715958760638046309173637, 12.58948648117584964655194266480, 13.52902894298763248881347149256, 15.41278414289167405492849730020, 16.87846261915840471198545124289

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