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Properties

Label 2-56-56.19-c1-0-2
Degree $2$
Conductor $56$
Sign $0.992 - 0.119i$
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  + (−1.30 + 0.554i)2-s + (1.18 − 0.686i)3-s + (1.38 − 1.44i)4-s + (−0.345 + 0.597i)5-s + (−1.16 + 1.55i)6-s + (2.63 − 0.222i)7-s + (−1.00 + 2.64i)8-s + (−0.557 + 0.966i)9-s + (0.117 − 0.969i)10-s + (−1.63 − 2.82i)11-s + (0.655 − 2.66i)12-s − 5.27·13-s + (−3.30 + 1.75i)14-s + 0.947i·15-s + (−0.167 − 3.99i)16-s + (−2.20 + 1.27i)17-s + ⋯
L(s)  = 1  + (−0.919 + 0.392i)2-s + (0.686 − 0.396i)3-s + (0.692 − 0.721i)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 + (0.0370 − 0.306i)10-s + (−0.491 − 0.851i)11-s + (0.189 − 0.769i)12-s − 1.46·13-s + (−0.883 + 0.468i)14-s + 0.244i·15-s + (−0.0417 − 0.999i)16-s + (−0.534 + 0.308i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.992 - 0.119i$
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: \((2,\ 56,\ (\ :1/2),\ 0.992 - 0.119i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.706553 + 0.0423044i\)
\(L(\frac12)\) \(\approx\) \(0.706553 + 0.0423044i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 - 0.554i)T \)
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

−15.09300841411871480116838465640, −14.55925575323521726998742826334, −13.33421125524781006014553226455, −11.51295025943200667929671446837, −10.65589682706757736359206790092, −9.092294682470265971436593326794, −8.010920681587108270976468400768, −7.27100069210191788075644195403, −5.33006556147870570644606568628, −2.38061587884815147924623858364, 2.52718474427026652349664480756, 4.62309434804969609022274494241, 7.19365277907818517875798704033, 8.332878406384761507662674829933, 9.313828801958527518844327820912, 10.40442441714275438535893639897, 11.74655228151181588371243449059, 12.69073097466348950724284932218, 14.53889802051739694131292564651, 15.19022843159202595874284654181

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