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Properties

Label 2-28-28.11-c2-0-5
Degree $2$
Conductor $28$
Sign $-0.788 + 0.615i$
Analytic cond. $0.762944$
Root an. cond. $0.873467$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.51 − 1.30i)2-s + (−3.95 − 2.28i)3-s + (0.597 + 3.95i)4-s + (−2.62 − 4.54i)5-s + (3.01 + 8.60i)6-s + (5.86 − 3.81i)7-s + (4.25 − 6.77i)8-s + (5.90 + 10.2i)9-s + (−1.95 + 10.3i)10-s + (−1.91 − 1.10i)11-s + (6.66 − 16.9i)12-s − 3.29·13-s + (−13.8 − 1.86i)14-s + 23.9i·15-s + (−15.2 + 4.72i)16-s + (6.69 − 11.6i)17-s + ⋯
L(s)  = 1  + (−0.758 − 0.652i)2-s + (−1.31 − 0.760i)3-s + (0.149 + 0.988i)4-s + (−0.525 − 0.909i)5-s + (0.502 + 1.43i)6-s + (0.838 − 0.545i)7-s + (0.531 − 0.846i)8-s + (0.655 + 1.13i)9-s + (−0.195 + 1.03i)10-s + (−0.174 − 0.100i)11-s + (0.555 − 1.41i)12-s − 0.253·13-s + (−0.991 − 0.133i)14-s + 1.59i·15-s + (−0.955 + 0.295i)16-s + (0.394 − 0.682i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $-0.788 + 0.615i$
Analytic conductor: \(0.762944\)
Root analytic conductor: \(0.873467\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :1),\ -0.788 + 0.615i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.139811 - 0.406058i\)
\(L(\frac12)\) \(\approx\) \(0.139811 - 0.406058i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.51 + 1.30i)T \)
7 \( 1 + (-5.86 + 3.81i)T \)
good3 \( 1 + (3.95 + 2.28i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (2.62 + 4.54i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (1.91 + 1.10i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 3.29T + 169T^{2} \)
17 \( 1 + (-6.69 + 11.6i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.72 + 3.88i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (3.66 - 2.11i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 39.4T + 841T^{2} \)
31 \( 1 + (-17.2 - 9.93i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-12.8 - 22.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 55.0T + 1.68e3T^{2} \)
43 \( 1 + 78.2iT - 1.84e3T^{2} \)
47 \( 1 + (18.3 - 10.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (24.0 - 41.6i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-66.7 - 38.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-23.7 - 41.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (45.2 + 26.1i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 90.1iT - 5.04e3T^{2} \)
73 \( 1 + (20.7 - 36.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-114. + 65.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 11.8iT - 6.88e3T^{2} \)
89 \( 1 + (15.5 + 26.9i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 140.T + 9.40e3T^{2} \)
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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

−16.95462444550224700252703829235, −16.02633178502293180169785871645, −13.52069348054286973758601658316, −12.14703243840487657281044677744, −11.69409643310490558642361843084, −10.36926461826523943070170753603, −8.375363918645870552016169936007, −7.14531919164706982098264221370, −4.84418028513637030504844735767, −0.902243226011562403949799107448, 4.97887499028383937925241511668, 6.37590751721768189499145050713, 8.026770635869523338248039385683, 9.972570562672271793678636438203, 10.95792966057093304199434551322, 11.82371267456278878291097249139, 14.53497906408402646658584135903, 15.35291532554817057308017749832, 16.34097772387881447088524105113, 17.50565165041620206666570133636

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