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Properties

Label 1-1056-1056.59-r0-0-0
Degree $1$
Conductor $1056$
Sign $0.770 + 0.637i$
Analytic cond. $4.90404$
Root an. cond. $4.90404$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.891 + 0.453i)5-s + (0.587 + 0.809i)7-s + (0.891 + 0.453i)13-s + (0.309 − 0.951i)17-s + (0.987 − 0.156i)19-s − i·23-s + (0.587 − 0.809i)25-s + (−0.156 + 0.987i)29-s + (−0.309 − 0.951i)31-s + (−0.891 − 0.453i)35-s + (−0.987 − 0.156i)37-s + (0.587 − 0.809i)41-s + (0.707 + 0.707i)43-s + (0.809 + 0.587i)47-s + (−0.309 + 0.951i)49-s + ⋯
L(s)  = 1  + (−0.891 + 0.453i)5-s + (0.587 + 0.809i)7-s + (0.891 + 0.453i)13-s + (0.309 − 0.951i)17-s + (0.987 − 0.156i)19-s − i·23-s + (0.587 − 0.809i)25-s + (−0.156 + 0.987i)29-s + (−0.309 − 0.951i)31-s + (−0.891 − 0.453i)35-s + (−0.987 − 0.156i)37-s + (0.587 − 0.809i)41-s + (0.707 + 0.707i)43-s + (0.809 + 0.587i)47-s + (−0.309 + 0.951i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1056 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1056 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1056\)    =    \(2^{5} \cdot 3 \cdot 11\)
Sign: $0.770 + 0.637i$
Analytic conductor: \(4.90404\)
Root analytic conductor: \(4.90404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1056} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1056,\ (0:\ ),\ 0.770 + 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.361390445 + 0.4902937195i\)
\(L(\frac12)\) \(\approx\) \(1.361390445 + 0.4902937195i\)
\(L(1)\) \(\approx\) \(1.054999303 + 0.1780872166i\)
\(L(1)\) \(\approx\) \(1.054999303 + 0.1780872166i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + (-0.891 + 0.453i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (0.891 + 0.453i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.987 - 0.156i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.156 + 0.987i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.987 - 0.156i)T \)
41 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.453 + 0.891i)T \)
59 \( 1 + (-0.987 - 0.156i)T \)
61 \( 1 + (0.453 + 0.891i)T \)
67 \( 1 + (0.707 - 0.707i)T \)
71 \( 1 + (0.951 + 0.309i)T \)
73 \( 1 + (0.587 + 0.809i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.453 - 0.891i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.309 + 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.16236812372161261461082093820, −20.657765310109224800142699221926, −19.88741567784381666570780343522, −19.28132889170776895478893173103, −18.29379905256921780352093735100, −17.45388987694705090147289890382, −16.786511530545829988161248942471, −15.84403327093890194389344991723, −15.356925879905673547280737847717, −14.29154345098435645624120515195, −13.56671636090826901119204003600, −12.70982113465802819272852682956, −11.83914479845783299799919983962, −11.10044529825011047806859974365, −10.41427542235339497594545333310, −9.32188974137649257973442498646, −8.26287837966798842872966196812, −7.83487523196022592180218609124, −6.97126941980355513211355268187, −5.729551844290634712531291283264, −4.91255517635662928623984462250, −3.82253943276053049180575030413, −3.43604572453445790141840991004, −1.67426990558424963215842663455, −0.83303907419763549455765893958, 0.98941135582316088524496725681, 2.33812720454918032479474811045, 3.211170851747314384307143927359, 4.19499787233641180479906507504, 5.11963769138617094405885866398, 6.06148205539437341876155914865, 7.113304271297056603975356182683, 7.79934937676981627534714998425, 8.729298421804458174871448012997, 9.3745327918769026307466995935, 10.72093170081563076353641888419, 11.27243622276141210712780032473, 11.98986317068556588232418723791, 12.69204963091195271873918143548, 14.091106258946261198106433279883, 14.35907244796154619253310213857, 15.59492906796868975070930380316, 15.82795819183575288336307375286, 16.811468478022110929533721749121, 18.0484012297306423556863580487, 18.50269805362095964298425640181, 19.033516454894492653274875991745, 20.20199642887237335568907822138, 20.70071749129959088507617076191, 21.62514996108606105601641834113

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