Properties

Label 1-56-56.11-r1-0-0
Degree $1$
Conductor $56$
Sign $-0.895 - 0.444i$
Analytic cond. $6.01803$
Root an. cond. $6.01803$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s − 13-s − 15-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s − 29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)33-s + (0.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s − 13-s − 15-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s − 29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)33-s + (0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $-0.895 - 0.444i$
Analytic conductor: \(6.01803\)
Root analytic conductor: \(6.01803\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 56,\ (1:\ ),\ -0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2127780276 - 0.9082936223i\)
\(L(\frac12)\) \(\approx\) \(0.2127780276 - 0.9082936223i\)
\(L(1)\) \(\approx\) \(0.6990897259 - 0.4650218252i\)
\(L(1)\) \(\approx\) \(0.6990897259 - 0.4650218252i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 - T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−33.44089760857108050533882887064, −32.31314427849658013068211840339, −31.02582026642803020475706988994, −29.71005535586695117929070605298, −28.71054655248816652230088619288, −27.60474158394939424828873065708, −26.388571914462508186307832720162, −25.72084177602388075666314901860, −23.94093297478523902248402235478, −22.66528232629425660293493084042, −21.908227826265944607319520075323, −20.86931957996837079506194929693, −19.38882017262599175102984463244, −17.78291035419521821393232715717, −17.12646220861935716819092767314, −15.376650802835246811951735895039, −14.76773452399755290162019886569, −13.01926126549215266048749945887, −11.391170592850916141809002293722, −10.326667724808013369870683634228, −9.36036697156193169166253895012, −7.24019005740766857875355064353, −5.81755931454230911076565056826, −4.38257876012493195080055360562, −2.561807585183171101102375643620, 0.526302844319604645236391096165, 2.31445603496412591753166182205, 4.91275094725652755567254160806, 6.06144687784873339162178666824, 7.63228666057880087569137432635, 8.983873631770688246855590808447, 10.69144534031462645991344060290, 12.16124496978078571724057788477, 13.06687760877605081775573291338, 14.2026002771023031059050468634, 16.234309791291067173955791344069, 17.086065714175798793766445583662, 18.25460211402449504551584536383, 19.4127170465784061773489912699, 20.74843583274657722691933261300, 22.024307826994827461401926752551, 23.31227544295145362654114420907, 24.51055603009769262657722873162, 24.958188249313275214275008582873, 26.67138757949227913581983410465, 28.04769879350391023623396315512, 29.168666954633157248086485137319, 29.6035910774765611977323694604, 31.23921960155871260564451031621, 32.13281654045398790134499577523

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