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Properties

Label 1-109-109.68-r1-0-0
Degree $1$
Conductor $109$
Sign $-0.622 + 0.782i$
Analytic cond. $11.7136$
Root an. cond. $11.7136$
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  + i·2-s + (−0.5 − 0.866i)3-s − 4-s + (−0.5 − 0.866i)5-s + (0.866 − 0.5i)6-s + (−0.5 − 0.866i)7-s − i·8-s + (−0.5 + 0.866i)9-s + (0.866 − 0.5i)10-s + (0.866 + 0.5i)11-s + (0.5 + 0.866i)12-s + (−0.866 + 0.5i)13-s + (0.866 − 0.5i)14-s + (−0.5 + 0.866i)15-s + 16-s + i·17-s + ⋯
L(s)  = 1  + i·2-s + (−0.5 − 0.866i)3-s − 4-s + (−0.5 − 0.866i)5-s + (0.866 − 0.5i)6-s + (−0.5 − 0.866i)7-s − i·8-s + (−0.5 + 0.866i)9-s + (0.866 − 0.5i)10-s + (0.866 + 0.5i)11-s + (0.5 + 0.866i)12-s + (−0.866 + 0.5i)13-s + (0.866 − 0.5i)14-s + (−0.5 + 0.866i)15-s + 16-s + i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(109\)
Sign: $-0.622 + 0.782i$
Analytic conductor: \(11.7136\)
Root analytic conductor: \(11.7136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{109} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 109,\ (1:\ ),\ -0.622 + 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1603262567 + 0.3322610497i\)
\(L(\frac12)\) \(\approx\) \(0.1603262567 + 0.3322610497i\)
\(L(1)\) \(\approx\) \(0.5952366728 + 0.08877286054i\)
\(L(1)\) \(\approx\) \(0.5952366728 + 0.08877286054i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad109 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + iT \)
19 \( 1 - iT \)
23 \( 1 + iT \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + iT \)
43 \( 1 - T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−29.014688138168146706039344150546, −27.7895310258179500044673921064, −27.14581691226464814692563503635, −26.42376842353465292247576920938, −24.882619895018424556312918224458, −23.07601298420896541880282845009, −22.44293336640051024347242588263, −21.91566044884606264082845630752, −20.72880391954017224257688428904, −19.519121556124068547626419160774, −18.6790819012931291852935120332, −17.56788622874295643133281053626, −16.251595723383036959101806043943, −15.01567545475068706339463626186, −14.13725026908094892582274883151, −12.17529632359231854811148308111, −11.77218310698332449641023284409, −10.48402721619951337956517533505, −9.68816508832553579802009060221, −8.46180139339157313195323103106, −6.41994098661113633098272355887, −5.05311604611472313127255307757, −3.65176139427585354565122936359, −2.72957264229909215447150913638, −0.19515068186398569508552024485, 1.20114720252530061139914542596, 4.022023090539582650623845327256, 5.14024580801813495750649348249, 6.65137402280852442252865003967, 7.32655657828836716369991586476, 8.551965807620420099114969265634, 9.84100528209109496142967754965, 11.69155946828850834197163598399, 12.77788216920304351758560185866, 13.545067103970231258041647799579, 14.86422731208234067114464907531, 16.28349115524188986382224009270, 17.041666648987582265437959560159, 17.64071797847935155306978614508, 19.41313118896975179479122178670, 19.69606096360152827915439636451, 21.80557784741830732889560099070, 22.89447453969604870765656414006, 23.72541987380581027986856072243, 24.31024062560561630741357482007, 25.33426746869805487653525301683, 26.423429571065521064550661604313, 27.66986639373132106346614740997, 28.40407366203979660789717922713, 29.6875006666465261114149002114

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