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Properties

Label 2-56-56.3-c1-0-0
Degree $2$
Conductor $56$
Sign $0.296 - 0.955i$
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 + (−0.416 − 0.240i)3-s + (−1.94 + 0.477i)4-s + (1.59 + 2.76i)5-s + (0.266 − 0.625i)6-s + (0.694 − 2.55i)7-s + (−0.999 − 2.64i)8-s + (−1.38 − 2.39i)9-s + (−3.61 + 2.71i)10-s + (0.800 − 1.38i)11-s + (0.923 + 0.268i)12-s − 1.38·13-s + (3.70 + 0.540i)14-s − 1.53i·15-s + (3.54 − 1.85i)16-s + (3.48 + 2.01i)17-s + ⋯
L(s)  = 1  + (0.120 + 0.992i)2-s + (−0.240 − 0.138i)3-s + (−0.971 + 0.238i)4-s + (0.714 + 1.23i)5-s + (0.108 − 0.255i)6-s + (0.262 − 0.964i)7-s + (−0.353 − 0.935i)8-s + (−0.461 − 0.799i)9-s + (−1.14 + 0.857i)10-s + (0.241 − 0.418i)11-s + (0.266 + 0.0774i)12-s − 0.385·13-s + (0.989 + 0.144i)14-s − 0.396i·15-s + (0.886 − 0.463i)16-s + (0.845 + 0.488i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.296 - 0.955i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1/2),\ 0.296 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.679297 + 0.500424i\)
\(L(\frac12)\) \(\approx\) \(0.679297 + 0.500424i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.694 + 2.55i)T \)
good3 \( 1 + (0.416 + 0.240i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.59 - 2.76i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.800 + 1.38i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.38T + 13T^{2} \)
17 \( 1 + (-3.48 - 2.01i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.56 - 2.63i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.83 - 2.21i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.10iT - 29T^{2} \)
31 \( 1 + (0.0579 - 0.100i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.63 - 2.67i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.21iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-5.05 - 8.76i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.13 - 3.54i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.38 + 2.53i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.21 + 7.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.01 + 8.69i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 + (-9.30 - 5.37i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.3 + 5.96i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.9iT - 83T^{2} \)
89 \( 1 + (-1.5 + 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.87iT - 97T^{2} \)
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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

−15.26471919884032658692676230840, −14.32231904970114104722852578902, −13.86181221046588967478742979815, −12.31260370706290523786878010874, −10.70554870643721000316842646601, −9.673751035589436658578175764006, −7.977218556795374272567443228196, −6.69475817690909313554539557738, −5.88071359817964704063112862272, −3.71851100473250361583582482992, 2.15983832251087429689402678176, 4.78194298745032568164569571545, 5.57637555101040971646264820839, 8.409918306769938684211647501948, 9.254000141675969425264204001226, 10.45007910362127310451250615853, 11.87410335070880401719929512203, 12.56942797711840506829046901143, 13.66817660180281782305695386624, 14.78446360208411867330720473170

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