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Properties

Label 2-28-28.19-c3-0-6
Degree $2$
Conductor $28$
Sign $-0.145 + 0.989i$
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  + (−1.75 + 2.21i)2-s + (−3.44 − 5.96i)3-s + (−1.81 − 7.79i)4-s + (−4.17 − 2.41i)5-s + (19.2 + 2.85i)6-s + (−5.03 − 17.8i)7-s + (20.4 + 9.67i)8-s + (−10.1 + 17.6i)9-s + (12.6 − 5.01i)10-s + (−36.6 + 21.1i)11-s + (−40.1 + 37.6i)12-s − 3.39i·13-s + (48.3 + 20.1i)14-s + 33.1i·15-s + (−57.4 + 28.2i)16-s + (101. − 58.6i)17-s + ⋯
L(s)  = 1  + (−0.621 + 0.783i)2-s + (−0.662 − 1.14i)3-s + (−0.227 − 0.973i)4-s + (−0.373 − 0.215i)5-s + (1.31 + 0.194i)6-s + (−0.272 − 0.962i)7-s + (0.903 + 0.427i)8-s + (−0.377 + 0.653i)9-s + (0.401 − 0.158i)10-s + (−1.00 + 0.579i)11-s + (−0.966 + 0.905i)12-s − 0.0724i·13-s + (0.922 + 0.385i)14-s + 0.571i·15-s + (−0.896 + 0.442i)16-s + (1.45 − 0.837i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $-0.145 + 0.989i$
Analytic conductor: \(1.65205\)
Root analytic conductor: \(1.28532\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :3/2),\ -0.145 + 0.989i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.344141 - 0.398365i\)
\(L(\frac12)\) \(\approx\) \(0.344141 - 0.398365i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.75 - 2.21i)T \)
7 \( 1 + (5.03 + 17.8i)T \)
good3 \( 1 + (3.44 + 5.96i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (4.17 + 2.41i)T + (62.5 + 108. i)T^{2} \)
11 \( 1 + (36.6 - 21.1i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 3.39iT - 2.19e3T^{2} \)
17 \( 1 + (-101. + 58.6i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-45.5 + 78.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-147. - 85.3i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 131.T + 2.43e4T^{2} \)
31 \( 1 + (-5.70 - 9.87i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-59.2 + 102. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 109. iT - 6.89e4T^{2} \)
43 \( 1 + 82.5iT - 7.95e4T^{2} \)
47 \( 1 + (36.7 - 63.6i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-87.2 - 151. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (166. + 288. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-472. - 272. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-516. + 298. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 384. iT - 3.57e5T^{2} \)
73 \( 1 + (-187. + 108. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-868. - 501. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 459.T + 5.71e5T^{2} \)
89 \( 1 + (771. + 445. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 282. iT - 9.12e5T^{2} \)
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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

−16.66090880278139353717377629651, −15.51896813647346939830191806079, −13.84958797959523213306673928025, −12.79329697569113844914251686732, −11.23405589865359834891578816763, −9.744848240467669435481798420280, −7.65721701120609619698717785046, −7.10339578309585769637737070374, −5.31463744927972684246645694688, −0.69573783523621706237239359132, 3.39341462708832315047572350899, 5.42060405145251106521295249203, 8.058036568567065420445067343246, 9.597085680958902652942478299635, 10.61378438341777872879413515167, 11.62689784517232674193305890464, 12.86388487013652917903452975426, 15.00993336321250934082767426176, 16.18192043747860157805657567695, 16.88713470804962148212964341840

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