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Properties

Label 2-56-56.19-c1-0-3
Degree $2$
Conductor $56$
Sign $0.665 - 0.746i$
Analytic cond. $0.447162$
Root an. cond. $0.668701$
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  + (0.169 + 1.40i)2-s + (1.18 − 0.686i)3-s + (−1.94 + 0.477i)4-s + (0.345 − 0.597i)5-s + (1.16 + 1.55i)6-s + (−2.63 + 0.222i)7-s + (−0.999 − 2.64i)8-s + (−0.557 + 0.966i)9-s + (0.897 + 0.382i)10-s + (−1.63 − 2.82i)11-s + (−1.98 + 1.90i)12-s + 5.27·13-s + (−0.760 − 3.66i)14-s − 0.947i·15-s + (3.54 − 1.85i)16-s + (−2.20 + 1.27i)17-s + ⋯
L(s)  = 1  + (0.120 + 0.992i)2-s + (0.686 − 0.396i)3-s + (−0.971 + 0.238i)4-s + (0.154 − 0.267i)5-s + (0.475 + 0.633i)6-s + (−0.996 + 0.0840i)7-s + (−0.353 − 0.935i)8-s + (−0.185 + 0.322i)9-s + (0.283 + 0.121i)10-s + (−0.491 − 0.851i)11-s + (−0.571 + 0.548i)12-s + 1.46·13-s + (−0.203 − 0.979i)14-s − 0.244i·15-s + (0.886 − 0.463i)16-s + (−0.534 + 0.308i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.665 - 0.746i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1/2),\ 0.665 - 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.859337 + 0.384963i\)
\(L(\frac12)\) \(\approx\) \(0.859337 + 0.384963i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.169 - 1.40i)T \)
7 \( 1 + (2.63 - 0.222i)T \)
good3 \( 1 + (-1.18 + 0.686i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.345 + 0.597i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.63 + 2.82i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.27T + 13T^{2} \)
17 \( 1 + (2.20 - 1.27i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.484 + 0.279i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.50 + 1.44i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.444iT - 29T^{2} \)
31 \( 1 + (-4.45 - 7.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.00 + 3.46i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.76iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-2.20 + 3.81i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.17 - 4.71i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.59 + 4.96i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.23 + 9.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.45 + 2.51i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 + (5.28 - 3.05i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.01 + 2.89i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.83iT - 83T^{2} \)
89 \( 1 + (-1.5 - 0.866i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.42iT - 97T^{2} \)
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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

−15.69591736119735782866625043381, −14.10637057087353542890619285575, −13.47173412755615716005114163891, −12.69451248603949406803252179252, −10.63818594735537234250710210889, −8.925271795247871797711437957992, −8.369549756563727723501829079463, −6.80784511139456711332113562838, −5.58242304557066225359821779201, −3.44036712928723524042188301584, 2.81034260296741703363706522537, 4.12153505613780386397733972440, 6.23169198841974162557561122863, 8.398667360432758101486844780789, 9.534523886484477441214522147149, 10.31589253902157197822706297561, 11.69919762468677485333046394495, 13.01744369340792731642545351791, 13.76572265924240420982959273118, 14.98404318640982501447181513767

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