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Properties

Label 2-28-28.3-c1-0-1
Degree $2$
Conductor $28$
Sign $0.832 + 0.553i$
Analytic cond. $0.223581$
Root an. cond. $0.472843$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 − 1.36i)2-s + (−0.866 + 1.5i)3-s + (−1.73 − i)4-s + (−1.5 + 0.866i)5-s + (1.73 + 1.73i)6-s + (1.73 − 2i)7-s + (−2 + 1.99i)8-s + (0.633 + 2.36i)10-s + (−0.866 − 0.5i)11-s + (3 − 1.73i)12-s − 3.46i·13-s + (−2.09 − 3.09i)14-s − 3i·15-s + (1.99 + 3.46i)16-s + (−1.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.499 + 0.866i)3-s + (−0.866 − 0.5i)4-s + (−0.670 + 0.387i)5-s + (0.707 + 0.707i)6-s + (0.654 − 0.755i)7-s + (−0.707 + 0.707i)8-s + (0.200 + 0.748i)10-s + (−0.261 − 0.150i)11-s + (0.866 − 0.499i)12-s − 0.960i·13-s + (−0.560 − 0.827i)14-s − 0.774i·15-s + (0.499 + 0.866i)16-s + (−0.363 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(0.223581\)
Root analytic conductor: \(0.472843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :1/2),\ 0.832 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.595521 - 0.179974i\)
\(L(\frac12)\) \(\approx\) \(0.595521 - 0.179974i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
7 \( 1 + (-1.73 + 2i)T \)
good3 \( 1 + (0.866 - 1.5i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (1.5 + 0.866i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.59 - 4.5i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (-0.866 + 1.5i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + (4.33 + 7.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.59 + 4.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.59 + 1.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 14iT - 71T^{2} \)
73 \( 1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.79 - 4.5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + (-13.5 + 7.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.3iT - 97T^{2} \)
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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

−17.32693724329760583976165417435, −15.87819465554784721739364562836, −14.71893569724865312437462693754, −13.40630963850504944178987305132, −11.76458414434093393874729517241, −10.81906248416036932328431871622, −10.02830839705813508541350777870, −7.948113451221559700796396614690, −5.21055972755172199076251286197, −3.79023935525388777841933541037, 4.72839043081946664752190802459, 6.41992193808807162362530336494, 7.74188899548957632199770281072, 9.025421267076234067385984904885, 11.67827119083028944990220808552, 12.43600940000912869275370860664, 13.76327518727332126738488530270, 15.18593175098594480682106124261, 16.10913315156939780936170302631, 17.52777971902752659332508962174

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