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Properties

Label 2-56-8.3-c2-0-9
Degree $2$
Conductor $56$
Sign $-0.467 + 0.883i$
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  + (−0.707 − 1.87i)2-s + 0.585·3-s + (−3 + 2.64i)4-s − 9.03i·5-s + (−0.414 − 1.09i)6-s − 2.64i·7-s + (7.07 + 3.74i)8-s − 8.65·9-s + (−16.8 + 6.38i)10-s + 12.4·11-s + (−1.75 + 1.54i)12-s + 9.03i·13-s + (−4.94 + 1.87i)14-s − 5.29i·15-s + (1.99 − 15.8i)16-s + 12.3·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.935i)2-s + 0.195·3-s + (−0.750 + 0.661i)4-s − 1.80i·5-s + (−0.0690 − 0.182i)6-s − 0.377i·7-s + (0.883 + 0.467i)8-s − 0.961·9-s + (−1.68 + 0.638i)10-s + 1.13·11-s + (−0.146 + 0.129i)12-s + 0.694i·13-s + (−0.353 + 0.133i)14-s − 0.352i·15-s + (0.124 − 0.992i)16-s + 0.726·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 + 0.883i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.467 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $-0.467 + 0.883i$
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: \((2,\ 56,\ (\ :1),\ -0.467 + 0.883i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.493809 - 0.819981i\)
\(L(\frac12)\) \(\approx\) \(0.493809 - 0.819981i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.87i)T \)
7 \( 1 + 2.64iT \)
good3 \( 1 - 0.585T + 9T^{2} \)
5 \( 1 + 9.03iT - 25T^{2} \)
11 \( 1 - 12.4T + 121T^{2} \)
13 \( 1 - 9.03iT - 169T^{2} \)
17 \( 1 - 12.3T + 289T^{2} \)
19 \( 1 - 28.8T + 361T^{2} \)
23 \( 1 - 24.6iT - 529T^{2} \)
29 \( 1 + 22.4iT - 841T^{2} \)
31 \( 1 - 16.7iT - 961T^{2} \)
37 \( 1 + 16.2iT - 1.36e3T^{2} \)
41 \( 1 - 6.97T + 1.68e3T^{2} \)
43 \( 1 + 22.8T + 1.84e3T^{2} \)
47 \( 1 - 6.19iT - 2.20e3T^{2} \)
53 \( 1 + 8.01iT - 2.80e3T^{2} \)
59 \( 1 - 30.4T + 3.48e3T^{2} \)
61 \( 1 - 15.2iT - 3.72e3T^{2} \)
67 \( 1 + 78.6T + 4.48e3T^{2} \)
71 \( 1 - 17.5iT - 5.04e3T^{2} \)
73 \( 1 - 46.6T + 5.32e3T^{2} \)
79 \( 1 - 81.0iT - 6.24e3T^{2} \)
83 \( 1 - 40.3T + 6.88e3T^{2} \)
89 \( 1 - 111.T + 7.92e3T^{2} \)
97 \( 1 + 164.T + 9.40e3T^{2} \)
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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

−14.16250316028913705899356532871, −13.42972614284200442306437354561, −12.05724792455478934182455677597, −11.60576222801396371579927001457, −9.600681078264537917804420000544, −9.018874129516209758765339489780, −7.86500541495313464296568637713, −5.28233043557842034962461835435, −3.80996221973949460325516762239, −1.22868638162074813288716917367, 3.23240569433970002948824000930, 5.73011218858143289333582397891, 6.79497190947018925674907188674, 7.978816107335857315460402053865, 9.409671001367252623054267783372, 10.56353196881501414121192179246, 11.77867912606977114174670939334, 13.82608899227714840246961899413, 14.55092935493162531535627368696, 15.06133719288301362135077362902

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