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Properties

Label 2-14e2-28.19-c1-0-4
Degree $2$
Conductor $196$
Sign $-0.133 - 0.991i$
Analytic cond. $1.56506$
Root an. cond. $1.25102$
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  + (1.10 + 0.878i)2-s + (1.07 + 1.85i)3-s + (0.455 + 1.94i)4-s + (−2.26 − 1.30i)5-s + (−0.443 + 2.99i)6-s + (−1.20 + 2.55i)8-s + (−0.792 + 1.37i)9-s + (−1.35 − 3.43i)10-s + (3.42 − 1.97i)11-s + (−3.12 + 2.92i)12-s + 1.08i·13-s − 5.59i·15-s + (−3.58 + 1.77i)16-s + (0.274 − 0.158i)17-s + (−2.08 + 0.824i)18-s + (2.58 − 4.47i)19-s + ⋯
L(s)  = 1  + (0.783 + 0.621i)2-s + (0.618 + 1.07i)3-s + (0.227 + 0.973i)4-s + (−1.01 − 0.584i)5-s + (−0.181 + 1.22i)6-s + (−0.426 + 0.904i)8-s + (−0.264 + 0.457i)9-s + (−0.429 − 1.08i)10-s + (1.03 − 0.596i)11-s + (−0.901 + 0.845i)12-s + 0.300i·13-s − 1.44i·15-s + (−0.896 + 0.443i)16-s + (0.0665 − 0.0384i)17-s + (−0.491 + 0.194i)18-s + (0.593 − 1.02i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.133 - 0.991i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1/2),\ -0.133 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22457 + 1.40048i\)
\(L(\frac12)\) \(\approx\) \(1.22457 + 1.40048i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.10 - 0.878i)T \)
7 \( 1 \)
good3 \( 1 + (-1.07 - 1.85i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.26 + 1.30i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.42 + 1.97i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.08iT - 13T^{2} \)
17 \( 1 + (-0.274 + 0.158i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.58 + 4.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.00 + 1.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
31 \( 1 + (-3.02 - 5.24i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.29iT - 41T^{2} \)
43 \( 1 - 7.23iT - 43T^{2} \)
47 \( 1 + (2.14 - 3.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.24 + 9.08i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.61 + 9.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.68 + 2.70i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.83 - 1.63i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (12.1 - 7.00i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.85 - 3.95i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.45T + 83T^{2} \)
89 \( 1 + (5.18 + 2.99i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.23iT - 97T^{2} \)
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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

−12.83937155703447349804743904748, −11.83508370163139924756099993067, −11.12431160864442960679821451768, −9.416416326692200116522696663783, −8.740005803709889590386164894882, −7.78304860586779784258521866738, −6.49494019150746136567176710808, −4.93322457368643223686847759513, −4.09494805935547742605016884963, −3.27021709284832573606934767473, 1.66617810164384891274543068218, 3.17412337929024668820743868137, 4.21238789730523935721223466099, 6.01605530344984476118888732464, 7.16861406142280546703927819155, 7.84074234882238186549015865497, 9.348471755817962892673759386078, 10.53015269391606315372805826105, 11.77598348759221802167530295676, 12.09288349840183625063466199329

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