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Properties

Label 2-56-7.6-c6-0-2
Degree $2$
Conductor $56$
Sign $-0.976 + 0.216i$
Analytic cond. $12.8830$
Root an. cond. $3.58929$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 39.5i·3-s + 231. i·5-s + (−334. + 74.2i)7-s − 834.·9-s + 2.30e3·11-s − 1.74e3i·13-s − 9.16e3·15-s − 3.67e3i·17-s + 6.69e3i·19-s + (−2.93e3 − 1.32e4i)21-s + 4.26e3·23-s − 3.80e4·25-s − 4.17e3i·27-s + 322.·29-s + 1.56e4i·31-s + ⋯
L(s)  = 1  + 1.46i·3-s + 1.85i·5-s + (−0.976 + 0.216i)7-s − 1.14·9-s + 1.72·11-s − 0.793i·13-s − 2.71·15-s − 0.747i·17-s + 0.975i·19-s + (−0.317 − 1.42i)21-s + 0.350·23-s − 2.43·25-s − 0.212i·27-s + 0.0132·29-s + 0.525i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $-0.976 + 0.216i$
Analytic conductor: \(12.8830\)
Root analytic conductor: \(3.58929\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :3),\ -0.976 + 0.216i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.153458 - 1.40057i\)
\(L(\frac12)\) \(\approx\) \(0.153458 - 1.40057i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (334. - 74.2i)T \)
good3 \( 1 - 39.5iT - 729T^{2} \)
5 \( 1 - 231. iT - 1.56e4T^{2} \)
11 \( 1 - 2.30e3T + 1.77e6T^{2} \)
13 \( 1 + 1.74e3iT - 4.82e6T^{2} \)
17 \( 1 + 3.67e3iT - 2.41e7T^{2} \)
19 \( 1 - 6.69e3iT - 4.70e7T^{2} \)
23 \( 1 - 4.26e3T + 1.48e8T^{2} \)
29 \( 1 - 322.T + 5.94e8T^{2} \)
31 \( 1 - 1.56e4iT - 8.87e8T^{2} \)
37 \( 1 + 5.78e4T + 2.56e9T^{2} \)
41 \( 1 - 1.07e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.33e5T + 6.32e9T^{2} \)
47 \( 1 - 242. iT - 1.07e10T^{2} \)
53 \( 1 - 9.55e3T + 2.21e10T^{2} \)
59 \( 1 - 3.80e4iT - 4.21e10T^{2} \)
61 \( 1 - 1.80e5iT - 5.15e10T^{2} \)
67 \( 1 + 1.77e5T + 9.04e10T^{2} \)
71 \( 1 + 1.35e5T + 1.28e11T^{2} \)
73 \( 1 + 5.43e5iT - 1.51e11T^{2} \)
79 \( 1 + 2.99e5T + 2.43e11T^{2} \)
83 \( 1 - 5.54e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.01e6iT - 4.96e11T^{2} \)
97 \( 1 + 1.06e6iT - 8.32e11T^{2} \)
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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

−14.76312993167541622282920700945, −14.01186255724211026141263220129, −12.01604650048234087939736990827, −10.82526219763571447022366144202, −10.06056816198471461686259635194, −9.180053861155962733477144400227, −7.05484903372424617780993501163, −5.92930291728104953210745201635, −3.83453677749452166045588305832, −3.04416035226329094625639691863, 0.63018106623042207816704126367, 1.63717370660406887731541132272, 4.18388751272919548109903067663, 6.06744505123651967426322647850, 7.10871933516278098505664021251, 8.677225747496813507317923069713, 9.354722741585736191491852270973, 11.70052647514632238296827206922, 12.47951335718275371520371607652, 13.14153793435852110773752153048

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