Properties

Label 2-56-56.5-c2-0-8
Degree $2$
Conductor $56$
Sign $0.797 + 0.603i$
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  + (0.611 − 1.90i)2-s + (1.93 + 3.35i)3-s + (−3.25 − 2.32i)4-s + (2.33 − 4.05i)5-s + (7.56 − 1.63i)6-s + (6.95 + 0.792i)7-s + (−6.42 + 4.77i)8-s + (−2.98 + 5.17i)9-s + (−6.28 − 6.92i)10-s + (−12.6 + 7.29i)11-s + (1.50 − 15.4i)12-s − 12.7·13-s + (5.75 − 12.7i)14-s + 18.1·15-s + (5.16 + 15.1i)16-s + (−16.9 + 9.76i)17-s + ⋯
L(s)  = 1  + (0.305 − 0.952i)2-s + (0.644 + 1.11i)3-s + (−0.813 − 0.581i)4-s + (0.467 − 0.810i)5-s + (1.26 − 0.272i)6-s + (0.993 + 0.113i)7-s + (−0.802 + 0.596i)8-s + (−0.331 + 0.575i)9-s + (−0.628 − 0.692i)10-s + (−1.14 + 0.663i)11-s + (0.125 − 1.28i)12-s − 0.977·13-s + (0.411 − 0.911i)14-s + 1.20·15-s + (0.322 + 0.946i)16-s + (−0.994 + 0.574i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.797 + 0.603i$
Analytic conductor: \(1.52588\)
Root analytic conductor: \(1.23526\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1),\ 0.797 + 0.603i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.46765 - 0.493119i\)
\(L(\frac12)\) \(\approx\) \(1.46765 - 0.493119i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.611 + 1.90i)T \)
7 \( 1 + (-6.95 - 0.792i)T \)
good3 \( 1 + (-1.93 - 3.35i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-2.33 + 4.05i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (12.6 - 7.29i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 12.7T + 169T^{2} \)
17 \( 1 + (16.9 - 9.76i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-8.86 + 15.3i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (4.43 - 7.67i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 35.4iT - 841T^{2} \)
31 \( 1 + (-25.1 + 14.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (10.5 + 6.10i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 22.0iT - 1.68e3T^{2} \)
43 \( 1 - 79.8iT - 1.84e3T^{2} \)
47 \( 1 + (36.5 + 21.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-31.3 + 18.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (1.20 + 2.08i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-14.6 + 25.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-35.2 + 20.3i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 22.6T + 5.04e3T^{2} \)
73 \( 1 + (-66.1 + 38.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (68.4 - 118. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 49.9T + 6.88e3T^{2} \)
89 \( 1 + (-0.970 - 0.560i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 158. iT - 9.40e3T^{2} \)
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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

−14.92532161704333758071320891355, −13.70447010241512595183392574377, −12.72429263668960012445674967018, −11.34659419765937548887905300812, −10.10876527694667641434433313960, −9.359770717168643355597713945333, −8.212785035217223529779167443901, −5.13000805422785768208597556791, −4.47061678662972138381964813757, −2.36876190465244836850949386906, 2.65047938537469230677115204304, 5.13794532771731567450992336956, 6.76349790871212719664649530279, 7.66157295009032853384743147814, 8.588407671376804688012036752781, 10.43146425699188846598803220079, 12.14593258225810182144483573607, 13.39475185386683192231200506367, 14.06151839924229663143132248145, 14.73850305389190242242600725714

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