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Properties

Label 2-28-28.3-c3-0-0
Degree $2$
Conductor $28$
Sign $-0.976 - 0.215i$
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  + (−2.26 + 1.69i)2-s + (−1.67 + 2.89i)3-s + (2.26 − 7.67i)4-s + (−15.9 + 9.21i)5-s + (−1.11 − 9.38i)6-s + (−15.4 − 10.1i)7-s + (7.87 + 21.2i)8-s + (7.91 + 13.7i)9-s + (20.5 − 47.9i)10-s + (35.6 + 20.5i)11-s + (18.4 + 19.3i)12-s + 24.8i·13-s + (52.2 − 3.21i)14-s − 61.5i·15-s + (−53.7 − 34.7i)16-s + (−41.3 − 23.8i)17-s + ⋯
L(s)  = 1  + (−0.800 + 0.598i)2-s + (−0.321 + 0.557i)3-s + (0.282 − 0.959i)4-s + (−1.42 + 0.824i)5-s + (−0.0760 − 0.638i)6-s + (−0.836 − 0.548i)7-s + (0.348 + 0.937i)8-s + (0.293 + 0.507i)9-s + (0.649 − 1.51i)10-s + (0.976 + 0.563i)11-s + (0.443 + 0.466i)12-s + 0.530i·13-s + (0.998 − 0.0613i)14-s − 1.06i·15-s + (−0.840 − 0.542i)16-s + (−0.590 − 0.340i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.215i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $-0.976 - 0.215i$
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: \((2,\ 28,\ (\ :3/2),\ -0.976 - 0.215i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0442710 + 0.406492i\)
\(L(\frac12)\) \(\approx\) \(0.0442710 + 0.406492i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.26 - 1.69i)T \)
7 \( 1 + (15.4 + 10.1i)T \)
good3 \( 1 + (1.67 - 2.89i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (15.9 - 9.21i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-35.6 - 20.5i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 24.8iT - 2.19e3T^{2} \)
17 \( 1 + (41.3 + 23.8i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-30.9 - 53.5i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (64.3 - 37.1i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 28.8T + 2.43e4T^{2} \)
31 \( 1 + (-11.5 + 20.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-51.8 - 89.8i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 96.1iT - 6.89e4T^{2} \)
43 \( 1 + 195. iT - 7.95e4T^{2} \)
47 \( 1 + (-89.8 - 155. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (218. - 378. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (286. - 496. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-368. + 212. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-728. - 420. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.17e3iT - 3.57e5T^{2} \)
73 \( 1 + (-716. - 413. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (282. - 162. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 507.T + 5.71e5T^{2} \)
89 \( 1 + (176. - 102. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.11e3iT - 9.12e5T^{2} \)
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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

−16.99044056348578452372672471799, −16.06853053869985706231615762113, −15.37047185081510317423736725094, −14.08234121350120386221787556513, −11.78555748727338415721983907441, −10.70301231828489268368593580780, −9.567045458413643778437526836144, −7.66836184287021825721141255714, −6.66712256001083697209664699904, −4.16911957932451292786993109675, 0.53876463615883275858356734982, 3.74524585294560984738940910770, 6.71900767638470376030078071661, 8.262620231138572214474546951027, 9.359769497984868036918601393259, 11.36053357612145169045322292899, 12.20926787758730877846281043768, 12.93437848643176512322863193148, 15.49918363561360471363143061068, 16.28142385107397805308646758181

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