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Properties

Label 2-28-28.3-c3-0-2
Degree $2$
Conductor $28$
Sign $-0.505 - 0.862i$
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  + (0.448 + 2.79i)2-s + (−2.11 + 3.65i)3-s + (−7.59 + 2.50i)4-s + (1.03 − 0.596i)5-s + (−11.1 − 4.25i)6-s + (16.7 + 7.86i)7-s + (−10.4 − 20.0i)8-s + (4.58 + 7.93i)9-s + (2.12 + 2.61i)10-s + (29.6 + 17.0i)11-s + (6.88 − 33.0i)12-s − 56.1i·13-s + (−14.4 + 50.3i)14-s + 5.04i·15-s + (51.4 − 38.0i)16-s + (19.4 + 11.2i)17-s + ⋯
L(s)  = 1  + (0.158 + 0.987i)2-s + (−0.406 + 0.703i)3-s + (−0.949 + 0.313i)4-s + (0.0924 − 0.0533i)5-s + (−0.759 − 0.289i)6-s + (0.905 + 0.424i)7-s + (−0.459 − 0.888i)8-s + (0.169 + 0.293i)9-s + (0.0673 + 0.0828i)10-s + (0.811 + 0.468i)11-s + (0.165 − 0.795i)12-s − 1.19i·13-s + (−0.276 + 0.961i)14-s + 0.0867i·15-s + (0.803 − 0.594i)16-s + (0.277 + 0.160i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $-0.505 - 0.862i$
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: \((2,\ 28,\ (\ :3/2),\ -0.505 - 0.862i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.560708 + 0.978264i\)
\(L(\frac12)\) \(\approx\) \(0.560708 + 0.978264i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.448 - 2.79i)T \)
7 \( 1 + (-16.7 - 7.86i)T \)
good3 \( 1 + (2.11 - 3.65i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-1.03 + 0.596i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-29.6 - 17.0i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 56.1iT - 2.19e3T^{2} \)
17 \( 1 + (-19.4 - 11.2i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (53.4 + 92.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (51.2 - 29.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 211.T + 2.43e4T^{2} \)
31 \( 1 + (-54.1 + 93.8i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (81.0 + 140. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 414. iT - 6.89e4T^{2} \)
43 \( 1 + 258. iT - 7.95e4T^{2} \)
47 \( 1 + (-304. - 527. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (112. - 194. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (42.9 - 74.3i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (312. - 180. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (580. + 334. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 133. iT - 3.57e5T^{2} \)
73 \( 1 + (675. + 389. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-729. + 421. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 432.T + 5.71e5T^{2} \)
89 \( 1 + (439. - 253. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 372. iT - 9.12e5T^{2} \)
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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

−17.23636202340162555870139167667, −15.69834371462399048204876755636, −15.09992937880766255494449733897, −13.75204429322326283686018795214, −12.23821597638918304936686039166, −10.57833397092464278918261771860, −9.078728882870460854551139552165, −7.61442985411791386695981617101, −5.67851403097531657272288113719, −4.45150682065665428792081799319, 1.44506950626293560690071940715, 4.24675341597283722864248394972, 6.37040472850861828237738539669, 8.401146444431987272936340282107, 10.07260862274737880441851222197, 11.57145782980472928182347434692, 12.18751929180135763456997628821, 13.77094698217265918671835420154, 14.52243062958873818798447759891, 16.76706062438346015820596493953

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