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Properties

Label 2-28-7.4-c5-0-0
Degree $2$
Conductor $28$
Sign $0.360 - 0.932i$
Analytic cond. $4.49074$
Root an. cond. $2.11913$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.77 + 3.08i)3-s + (20.8 + 36.0i)5-s + (28 + 126. i)7-s + (115. + 199. i)9-s + (−55.3 + 95.9i)11-s + 179.·13-s − 148.·15-s + (177. − 307. i)17-s + (−977. − 1.69e3i)19-s + (−440. − 138. i)21-s + (−773. − 1.33e3i)23-s + (695. − 1.20e3i)25-s − 1.68e3·27-s + 6.27e3·29-s + (−3.00e3 + 5.19e3i)31-s + ⋯
L(s)  = 1  + (−0.114 + 0.197i)3-s + (0.372 + 0.645i)5-s + (0.215 + 0.976i)7-s + (0.473 + 0.820i)9-s + (−0.137 + 0.239i)11-s + 0.293·13-s − 0.170·15-s + (0.149 − 0.258i)17-s + (−0.621 − 1.07i)19-s + (−0.217 − 0.0687i)21-s + (−0.304 − 0.527i)23-s + (0.222 − 0.385i)25-s − 0.444·27-s + 1.38·29-s + (−0.561 + 0.971i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.360 - 0.932i$
Analytic conductor: \(4.49074\)
Root analytic conductor: \(2.11913\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :5/2),\ 0.360 - 0.932i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.24361 + 0.852185i\)
\(L(\frac12)\) \(\approx\) \(1.24361 + 0.852185i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-28 - 126. i)T \)
good3 \( 1 + (1.77 - 3.08i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-20.8 - 36.0i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (55.3 - 95.9i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 179.T + 3.71e5T^{2} \)
17 \( 1 + (-177. + 307. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (977. + 1.69e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (773. + 1.33e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 6.27e3T + 2.05e7T^{2} \)
31 \( 1 + (3.00e3 - 5.19e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-4.84e3 - 8.39e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.05e4T + 1.15e8T^{2} \)
43 \( 1 - 6.71e3T + 1.47e8T^{2} \)
47 \( 1 + (1.36e4 + 2.35e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.63e4 + 2.83e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-246. + 426. i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (2.02e4 + 3.51e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-3.84e3 + 6.65e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 7.78e4T + 1.80e9T^{2} \)
73 \( 1 + (3.69e4 - 6.40e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-2.19e4 - 3.80e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 4.11e4T + 3.93e9T^{2} \)
89 \( 1 + (3.28e4 + 5.69e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 6.85e4T + 8.58e9T^{2} \)
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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

−16.26587562286421760951033856001, −15.24763567413731191987067776571, −14.04038865345788804277564430719, −12.67234883576040145500978059593, −11.17462790716574315087503796615, −10.03804765478070224401771948908, −8.433881689701672670447128717991, −6.63770532675173131210176822447, −4.94184237433482742391751810834, −2.42721347099846571737991193663, 1.13347534732436320568214951797, 4.09354277326655723022911140888, 6.06975733832410476113673053991, 7.74554255132664598056058465262, 9.375985378834124271067946470231, 10.72835068401309636092867991536, 12.34651563213238244945505234123, 13.35518198243849119326151603178, 14.61691588630095439822576404448, 16.15417943034381318989137509511

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