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Properties

Label 1-33e2-1089.79-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.336 + 0.941i$
Analytic cond. $117.029$
Root an. cond. $117.029$
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  + (0.969 + 0.244i)2-s + (0.879 + 0.475i)4-s + (0.861 − 0.508i)5-s + (−0.161 + 0.986i)7-s + (0.736 + 0.676i)8-s + (0.959 − 0.281i)10-s + (0.991 − 0.132i)13-s + (−0.398 + 0.917i)14-s + (0.548 + 0.836i)16-s + (−0.0855 + 0.996i)17-s + (0.921 − 0.389i)19-s + (0.999 − 0.0380i)20-s + (−0.888 + 0.458i)23-s + (0.483 − 0.875i)25-s + (0.993 + 0.113i)26-s + ⋯
L(s)  = 1  + (0.969 + 0.244i)2-s + (0.879 + 0.475i)4-s + (0.861 − 0.508i)5-s + (−0.161 + 0.986i)7-s + (0.736 + 0.676i)8-s + (0.959 − 0.281i)10-s + (0.991 − 0.132i)13-s + (−0.398 + 0.917i)14-s + (0.548 + 0.836i)16-s + (−0.0855 + 0.996i)17-s + (0.921 − 0.389i)19-s + (0.999 − 0.0380i)20-s + (−0.888 + 0.458i)23-s + (0.483 − 0.875i)25-s + (0.993 + 0.113i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.336 + 0.941i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.336 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.685395383 + 3.300933693i\)
\(L(\frac12)\) \(\approx\) \(4.685395383 + 3.300933693i\)
\(L(1)\) \(\approx\) \(2.341542001 + 0.7318929301i\)
\(L(1)\) \(\approx\) \(2.341542001 + 0.7318929301i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.969 + 0.244i)T \)
5 \( 1 + (0.861 - 0.508i)T \)
7 \( 1 + (-0.161 + 0.986i)T \)
13 \( 1 + (0.991 - 0.132i)T \)
17 \( 1 + (-0.0855 + 0.996i)T \)
19 \( 1 + (0.921 - 0.389i)T \)
23 \( 1 + (-0.888 + 0.458i)T \)
29 \( 1 + (0.999 + 0.0190i)T \)
31 \( 1 + (-0.761 + 0.647i)T \)
37 \( 1 + (-0.564 - 0.825i)T \)
41 \( 1 + (0.830 - 0.556i)T \)
43 \( 1 + (0.995 - 0.0950i)T \)
47 \( 1 + (0.345 + 0.938i)T \)
53 \( 1 + (-0.998 - 0.0570i)T \)
59 \( 1 + (-0.0665 + 0.997i)T \)
61 \( 1 + (-0.272 + 0.962i)T \)
67 \( 1 + (-0.786 - 0.618i)T \)
71 \( 1 + (0.974 + 0.226i)T \)
73 \( 1 + (0.254 - 0.967i)T \)
79 \( 1 + (0.948 - 0.318i)T \)
83 \( 1 + (-0.988 + 0.151i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (0.861 + 0.508i)T \)
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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

−21.01546691702384896441682715078, −20.527187030148161933757359502776, −19.880484116224889703000795292207, −18.72755348707150991004716206351, −18.143208291646437205370298784, −17.0891275618996037353574727985, −16.21239097923901304797129196038, −15.661331979044382072884583046028, −14.33055594751474707329787563948, −14.02510728076788072424414893707, −13.44590761849852506613037721061, −12.58945123253805496030828897797, −11.50919062185018861001444549262, −10.86711234653242941259988547792, −10.082425295776354101271122260048, −9.44961240698461601864212002669, −7.92915154853856212437002353237, −6.99589809878635461767407280197, −6.35160457730966525165459425887, −5.55783691193130785886413604965, −4.54960050149021843179722766215, −3.61446291674250193331900536898, −2.85066585983568790834095235436, −1.75995656096810554591691833042, −0.81218459376072009200526744193, 1.29207817638017355433804052013, 2.16194691543794335944517848207, 3.115545637718094429957712689127, 4.12115373851659437710680186413, 5.194382759513082326386569079993, 5.86466979210441691974012161335, 6.30691601559767968161131966474, 7.57167830514080810000486085320, 8.57119683380199284962053312335, 9.22370540434518941750111709708, 10.407868641426253405548904826189, 11.21019966746618330191106485641, 12.39378298542391099827520903560, 12.56674433291387792154457507710, 13.69774624219293806731092300124, 14.08205022223223065601227213167, 15.174983456182632332090846291355, 15.93261063402201593654484470763, 16.35415568589105505246727879308, 17.67046878842630094945100175489, 17.90917467465115956257624569947, 19.268950756493856873456099624899, 20.03467065304379739608807707866, 20.99863125934742606128437727422, 21.3900530369341550951000315529

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