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Properties

Label 1-33e2-1089.295-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.602 - 0.797i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
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  + (−0.683 − 0.730i)2-s + (−0.0665 + 0.997i)4-s + (−0.625 − 0.780i)5-s + (−0.851 + 0.524i)7-s + (0.774 − 0.633i)8-s + (−0.142 + 0.989i)10-s + (−0.532 − 0.846i)13-s + (0.964 + 0.263i)14-s + (−0.991 − 0.132i)16-s + (0.198 + 0.980i)17-s + (0.993 − 0.113i)19-s + (0.820 − 0.572i)20-s + (0.235 − 0.971i)23-s + (−0.217 + 0.976i)25-s + (−0.254 + 0.967i)26-s + ⋯
L(s)  = 1  + (−0.683 − 0.730i)2-s + (−0.0665 + 0.997i)4-s + (−0.625 − 0.780i)5-s + (−0.851 + 0.524i)7-s + (0.774 − 0.633i)8-s + (−0.142 + 0.989i)10-s + (−0.532 − 0.846i)13-s + (0.964 + 0.263i)14-s + (−0.991 − 0.132i)16-s + (0.198 + 0.980i)17-s + (0.993 − 0.113i)19-s + (0.820 − 0.572i)20-s + (0.235 − 0.971i)23-s + (−0.217 + 0.976i)25-s + (−0.254 + 0.967i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.602 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.602 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.602 - 0.797i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.602 - 0.797i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2531510011 - 0.5085687668i\)
\(L(\frac12)\) \(\approx\) \(0.2531510011 - 0.5085687668i\)
\(L(1)\) \(\approx\) \(0.5348773221 - 0.2511238732i\)
\(L(1)\) \(\approx\) \(0.5348773221 - 0.2511238732i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.683 - 0.730i)T \)
5 \( 1 + (-0.625 - 0.780i)T \)
7 \( 1 + (-0.851 + 0.524i)T \)
13 \( 1 + (-0.532 - 0.846i)T \)
17 \( 1 + (0.198 + 0.980i)T \)
19 \( 1 + (0.993 - 0.113i)T \)
23 \( 1 + (0.235 - 0.971i)T \)
29 \( 1 + (0.953 + 0.299i)T \)
31 \( 1 + (0.272 + 0.962i)T \)
37 \( 1 + (-0.985 + 0.170i)T \)
41 \( 1 + (-0.999 + 0.0190i)T \)
43 \( 1 + (0.0475 - 0.998i)T \)
47 \( 1 + (0.797 + 0.603i)T \)
53 \( 1 + (0.610 + 0.791i)T \)
59 \( 1 + (0.483 + 0.875i)T \)
61 \( 1 + (-0.290 - 0.956i)T \)
67 \( 1 + (-0.327 - 0.945i)T \)
71 \( 1 + (-0.870 - 0.491i)T \)
73 \( 1 + (-0.564 - 0.825i)T \)
79 \( 1 + (0.449 + 0.893i)T \)
83 \( 1 + (-0.761 - 0.647i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (-0.625 + 0.780i)T \)
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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

−22.017034132033526052324387801724, −20.68247382440730410289561595085, −19.799711384495304995422807006200, −19.243412137773385059103231417133, −18.68345899154352026013384884164, −17.82739363320660817748777314169, −16.98838043210215553754038258923, −16.12351707004722246790277630415, −15.74841022393249430929123184339, −14.79102389389459400232765641291, −13.99414770205261754366354496940, −13.443437909540153676671713710811, −11.88673220831732968726779971523, −11.428827540978336722501659480775, −10.19758343032379452561586014391, −9.82676458017448846815897701250, −8.91236894786762885067321480414, −7.745844024850111594908139928570, −7.14713041033777131222601915927, −6.667475511782184003481322448109, −5.5806648312494976865170383200, −4.50749590886399280786208781906, −3.46237576867863821494953174947, −2.430270420430331925090474861813, −0.92580159601652564005050382026, 0.39616447425816930575033088554, 1.51123290099382169986272306910, 2.84759539170308057915682318560, 3.42432541661712680919763282514, 4.54087415870620553352640200616, 5.48273630651559151078233640683, 6.77243908346175922284553013633, 7.66774179762616177404590603788, 8.575390128596752853636029620389, 8.99921191451634663861805253580, 10.11400729163404444801860746005, 10.58776048216175782731768897319, 11.964874715200122671780009240808, 12.26149288075888359081385743374, 12.882922759086338705662940278244, 13.83200869553141025475679174877, 15.26076014543992239866091305033, 15.81908477291684686129275442315, 16.62885774615607812844890853034, 17.28055189981626827547230110477, 18.19479952249111662204259358539, 19.03998824753433400850331100339, 19.63245894977128230265534624937, 20.18856624813818775152595470827, 20.94995011829463340613314561313

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