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
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: | $\chi_{109} (101, \cdot )$ |
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
$p$ | $F_p(T)$ | |
---|---|---|
bad | 109 | \( 1 \) |
good | 2 | \( 1 - iT \) |
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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Imaginary part of the first few zeros on the critical line
−29.6875006666465261114149002114, −28.40407366203979660789717922713, −27.66986639373132106346614740997, −26.423429571065521064550661604313, −25.33426746869805487653525301683, −24.31024062560561630741357482007, −23.72541987380581027986856072243, −22.89447453969604870765656414006, −21.80557784741830732889560099070, −19.69606096360152827915439636451, −19.41313118896975179479122178670, −17.64071797847935155306978614508, −17.041666648987582265437959560159, −16.28349115524188986382224009270, −14.86422731208234067114464907531, −13.545067103970231258041647799579, −12.77788216920304351758560185866, −11.69155946828850834197163598399, −9.84100528209109496142967754965, −8.551965807620420099114969265634, −7.32655657828836716369991586476, −6.65137402280852442252865003967, −5.14024580801813495750649348249, −4.022023090539582650623845327256, −1.20114720252530061139914542596, 0.19515068186398569508552024485, 2.72957264229909215447150913638, 3.65176139427585354565122936359, 5.05311604611472313127255307757, 6.41994098661113633098272355887, 8.46180139339157313195323103106, 9.68816508832553579802009060221, 10.48402721619951337956517533505, 11.77218310698332449641023284409, 12.17529632359231854811148308111, 14.13725026908094892582274883151, 15.01567545475068706339463626186, 16.251595723383036959101806043943, 17.56788622874295643133281053626, 18.6790819012931291852935120332, 19.519121556124068547626419160774, 20.72880391954017224257688428904, 21.91566044884606264082845630752, 22.44293336640051024347242588263, 23.07601298420896541880282845009, 24.882619895018424556312918224458, 26.42376842353465292247576920938, 27.14581691226464814692563503635, 27.7895310258179500044673921064, 29.014688138168146706039344150546