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Properties

Label 2-56-8.5-c1-0-0
Degree $2$
Conductor $56$
Sign $0.254 - 0.967i$
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  + (−1.28 + 0.599i)2-s + 0.936i·3-s + (1.28 − 1.53i)4-s + 3.33i·5-s + (−0.561 − 1.19i)6-s − 7-s + (−0.719 + 2.73i)8-s + 2.12·9-s + (−2 − 4.27i)10-s − 4.27i·11-s + (1.43 + 1.19i)12-s − 3.33i·13-s + (1.28 − 0.599i)14-s − 3.12·15-s + (−0.719 − 3.93i)16-s + 2·17-s + ⋯
L(s)  = 1  + (−0.905 + 0.424i)2-s + 0.540i·3-s + (0.640 − 0.768i)4-s + 1.49i·5-s + (−0.229 − 0.489i)6-s − 0.377·7-s + (−0.254 + 0.967i)8-s + 0.707·9-s + (−0.632 − 1.35i)10-s − 1.28i·11-s + (0.415 + 0.346i)12-s − 0.924i·13-s + (0.342 − 0.160i)14-s − 0.806·15-s + (−0.179 − 0.983i)16-s + 0.485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.254 - 0.967i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1/2),\ 0.254 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.478741 + 0.369138i\)
\(L(\frac12)\) \(\approx\) \(0.478741 + 0.369138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 - 0.599i)T \)
7 \( 1 + T \)
good3 \( 1 - 0.936iT - 3T^{2} \)
5 \( 1 - 3.33iT - 5T^{2} \)
11 \( 1 + 4.27iT - 11T^{2} \)
13 \( 1 + 3.33iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 0.936iT - 19T^{2} \)
23 \( 1 + 3.12T + 23T^{2} \)
29 \( 1 + 1.87iT - 29T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 - 1.87iT - 37T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 + 4.27iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 8.54iT - 53T^{2} \)
59 \( 1 - 7.60iT - 59T^{2} \)
61 \( 1 + 3.33iT - 61T^{2} \)
67 \( 1 - 15.7iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 9.47iT - 83T^{2} \)
89 \( 1 - 0.246T + 89T^{2} \)
97 \( 1 + 4.24T + 97T^{2} \)
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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

−15.58660838017371366833751761003, −14.81706691221366276457953394907, −13.65442537169822740716688386590, −11.61768538675554535396347976514, −10.39621828008299551358890428409, −10.04911255433410882859123356370, −8.313382666521679906312296326613, −7.01386485095758297775178366062, −5.85446107856585298320374740528, −3.19000730340936300109251659712, 1.62949650153129759522398357534, 4.44694036039398950736658268035, 6.75978578429511292260690995362, 7.969963113737546375805271881448, 9.236909328580011221848236932735, 10.04996230736045946420997935492, 12.00687881367141584837581783940, 12.44663567645430761503622998397, 13.45449540608430454882740680529, 15.50334710527529026660877584228

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