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Properties

Label 2-28-28.19-c1-0-0
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  + (−1.36 + 0.366i)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 + (−1.99 + 2i)8-s + (2.36 + 0.633i)10-s + (0.866 − 0.5i)11-s + (3 + 1.73i)12-s + 3.46i·13-s + (3.09 + 2.09i)14-s − 3i·15-s + (1.99 − 3.46i)16-s + (−1.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)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 + (0.748 + 0.200i)10-s + (0.261 − 0.150i)11-s + (0.866 + 0.499i)12-s + 0.960i·13-s + (0.827 + 0.560i)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: $\chi_{28} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :1/2),\ 0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.478059 + 0.144476i\)
\(L(\frac12)\) \(\approx\) \(0.478059 + 0.144476i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
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

−17.06857603559668187940785591779, −16.25244023919090663483888074103, −15.41511323957353834842249934522, −14.16209946289559761358393547268, −12.13207478256375503000632186598, −10.58044058794066070153863344420, −9.500752157084911352039298183007, −8.366268142254710935803300809034, −6.70339154631673365237492672570, −3.97935651422220664617584685071, 2.79175426214166254513393836923, 6.68917178271240966326952511997, 7.85822589789037405549931951894, 9.085135028380696945126021414053, 10.75381845164304346558910523698, 12.15519613932422405978079593033, 13.13342652700473590833995356274, 15.09420966833529296501254558014, 15.97201328306283097025206163750, 17.60497493703880247588484279251

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