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Properties

Label 2-56-8.3-c2-0-3
Degree $2$
Conductor $56$
Sign $0.986 + 0.163i$
Analytic cond. $1.52588$
Root an. cond. $1.23526$
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  + (−1.67 − 1.09i)2-s + 4.56·3-s + (1.60 + 3.66i)4-s + 5.73i·5-s + (−7.64 − 4.99i)6-s − 2.64i·7-s + (1.30 − 7.89i)8-s + 11.8·9-s + (6.26 − 9.60i)10-s − 1.40·11-s + (7.34 + 16.7i)12-s − 19.0i·13-s + (−2.89 + 4.43i)14-s + 26.1i·15-s + (−10.8 + 11.7i)16-s − 32.2·17-s + ⋯
L(s)  = 1  + (−0.837 − 0.546i)2-s + 1.52·3-s + (0.402 + 0.915i)4-s + 1.14i·5-s + (−1.27 − 0.832i)6-s − 0.377i·7-s + (0.163 − 0.986i)8-s + 1.31·9-s + (0.626 − 0.960i)10-s − 0.127·11-s + (0.612 + 1.39i)12-s − 1.46i·13-s + (−0.206 + 0.316i)14-s + 1.74i·15-s + (−0.676 + 0.736i)16-s − 1.89·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.986 + 0.163i$
Analytic conductor: \(1.52588\)
Root analytic conductor: \(1.23526\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1),\ 0.986 + 0.163i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.20011 - 0.0988798i\)
\(L(\frac12)\) \(\approx\) \(1.20011 - 0.0988798i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.67 + 1.09i)T \)
7 \( 1 + 2.64iT \)
good3 \( 1 - 4.56T + 9T^{2} \)
5 \( 1 - 5.73iT - 25T^{2} \)
11 \( 1 + 1.40T + 121T^{2} \)
13 \( 1 + 19.0iT - 169T^{2} \)
17 \( 1 + 32.2T + 289T^{2} \)
19 \( 1 - 12.5T + 361T^{2} \)
23 \( 1 - 15.8iT - 529T^{2} \)
29 \( 1 + 3.29iT - 841T^{2} \)
31 \( 1 - 22.6iT - 961T^{2} \)
37 \( 1 + 54.1iT - 1.36e3T^{2} \)
41 \( 1 + 7.59T + 1.68e3T^{2} \)
43 \( 1 + 20.8T + 1.84e3T^{2} \)
47 \( 1 - 21.6iT - 2.20e3T^{2} \)
53 \( 1 - 0.356iT - 2.80e3T^{2} \)
59 \( 1 - 26.8T + 3.48e3T^{2} \)
61 \( 1 - 86.2iT - 3.72e3T^{2} \)
67 \( 1 - 114.T + 4.48e3T^{2} \)
71 \( 1 + 104. iT - 5.04e3T^{2} \)
73 \( 1 + 24.3T + 5.32e3T^{2} \)
79 \( 1 - 117. iT - 6.24e3T^{2} \)
83 \( 1 - 79.2T + 6.88e3T^{2} \)
89 \( 1 - 2.66T + 7.92e3T^{2} \)
97 \( 1 + 52.0T + 9.40e3T^{2} \)
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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

−15.07604140286967705370310450798, −13.81892169411431566563600937308, −12.93117160453561281063392552581, −11.11879984597932630622308626471, −10.24297278883832008113414428255, −9.070624089077121888486930916912, −7.921955421990733648976060533628, −7.00060725884540152199769280461, −3.55482134082125469948507007129, −2.50383677884877753339977415568, 2.06084243562262991664560995984, 4.65311898925642541781413155323, 6.76813598874068242060133006886, 8.308586874490709147427466396782, 8.884248313745473803238352523566, 9.639213971637014990224690338835, 11.52457907317396200366002562921, 13.18651483339511941165757860693, 14.09931437782805059606454280636, 15.21424643824394719810019411148

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