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Properties

Label 2-28-7.3-c4-0-0
Degree $2$
Conductor $28$
Sign $0.719 - 0.694i$
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  + (13.6 + 7.89i)3-s + (−24.9 + 14.3i)5-s + (39.2 − 29.2i)7-s + (84.0 + 145. i)9-s + (6.39 − 11.0i)11-s − 283. i·13-s − 453.·15-s + (−164. − 94.7i)17-s + (−139. + 80.7i)19-s + (768. − 90.5i)21-s + (−86.2 − 149. i)23-s + (101. − 175. i)25-s + 1.37e3i·27-s + 711.·29-s + (−747. − 431. i)31-s + ⋯
L(s)  = 1  + (1.51 + 0.876i)3-s + (−0.996 + 0.575i)5-s + (0.801 − 0.597i)7-s + (1.03 + 1.79i)9-s + (0.0528 − 0.0915i)11-s − 1.67i·13-s − 2.01·15-s + (−0.567 − 0.327i)17-s + (−0.387 + 0.223i)19-s + (1.74 − 0.205i)21-s + (−0.163 − 0.282i)23-s + (0.161 − 0.280i)25-s + 1.88i·27-s + 0.845·29-s + (−0.777 − 0.449i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.719 - 0.694i$
Analytic conductor: \(2.89435\)
Root analytic conductor: \(1.70128\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :2),\ 0.719 - 0.694i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.74191 + 0.703516i\)
\(L(\frac12)\) \(\approx\) \(1.74191 + 0.703516i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-39.2 + 29.2i)T \)
good3 \( 1 + (-13.6 - 7.89i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (24.9 - 14.3i)T + (312.5 - 541. i)T^{2} \)
11 \( 1 + (-6.39 + 11.0i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + 283. iT - 2.85e4T^{2} \)
17 \( 1 + (164. + 94.7i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (139. - 80.7i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (86.2 + 149. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 - 711.T + 7.07e5T^{2} \)
31 \( 1 + (747. + 431. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (-411. - 712. i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 - 2.85e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.32e3T + 3.41e6T^{2} \)
47 \( 1 + (2.99e3 - 1.73e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (102. - 177. i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-3.35e3 - 1.93e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-1.78e3 + 1.03e3i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (63.1 - 109. i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 7.59e3T + 2.54e7T^{2} \)
73 \( 1 + (4.44e3 + 2.56e3i)T + (1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-1.35e3 - 2.35e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + 7.84e3iT - 4.74e7T^{2} \)
89 \( 1 + (5.59e3 - 3.23e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + 189. iT - 8.85e7T^{2} \)
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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

−16.13520341202990485291149170472, −15.09302003866162268489333647191, −14.58114211842843390371958185250, −13.22974559396061072357000603312, −11.15087537092366326977182015337, −10.10970535187527452147180841108, −8.374618511738670666522175852297, −7.63567424134319868402458843479, −4.41755758246452247969757970302, −3.07518507245500106957165677613, 1.95514984682086819476000089530, 4.16155882251799136389917218763, 7.08170355473327736673074111044, 8.371032269516250187399827606768, 9.008165381506086708610855388915, 11.61200782448187607653869573108, 12.60606461597066123825823799542, 13.96991305263443536991798424432, 14.85719029982085357157326037802, 16.00424460128009152907733081594

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