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Properties

Label 1-109-109.75-r0-0-0
Degree $1$
Conductor $109$
Sign $-0.985 - 0.170i$
Analytic cond. $0.506193$
Root an. cond. $0.506193$
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  + (−0.5 − 0.866i)2-s + (0.173 + 0.984i)3-s + (−0.5 + 0.866i)4-s + (−0.939 − 0.342i)5-s + (0.766 − 0.642i)6-s + (−0.939 − 0.342i)7-s + 8-s + (−0.939 + 0.342i)9-s + (0.173 + 0.984i)10-s + (0.173 − 0.984i)11-s + (−0.939 − 0.342i)12-s + (−0.939 − 0.342i)13-s + (0.173 + 0.984i)14-s + (0.173 − 0.984i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.173 + 0.984i)3-s + (−0.5 + 0.866i)4-s + (−0.939 − 0.342i)5-s + (0.766 − 0.642i)6-s + (−0.939 − 0.342i)7-s + 8-s + (−0.939 + 0.342i)9-s + (0.173 + 0.984i)10-s + (0.173 − 0.984i)11-s + (−0.939 − 0.342i)12-s + (−0.939 − 0.342i)13-s + (0.173 + 0.984i)14-s + (0.173 − 0.984i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(109\)
Sign: $-0.985 - 0.170i$
Analytic conductor: \(0.506193\)
Root analytic conductor: \(0.506193\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{109} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 109,\ (0:\ ),\ -0.985 - 0.170i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01378522216 - 0.1602862837i\)
\(L(\frac12)\) \(\approx\) \(0.01378522216 - 0.1602862837i\)
\(L(1)\) \(\approx\) \(0.4415917429 - 0.1293567560i\)
\(L(1)\) \(\approx\) \(0.4415917429 - 0.1293567560i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad109 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (0.173 - 0.984i)T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.766 + 0.642i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.173 + 0.984i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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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

−30.03048060136259171909203189476, −28.77595737310100863423147818790, −28.00512891705305025457515739177, −26.619463927998540478960999704475, −25.91365475784696773655427622198, −24.92235516708106537793972212177, −24.02688604499319794960505568915, −23.03177887543834731416053056900, −22.40525913883581261732713150310, −20.01826644418698918529580941585, −19.339492145006253387321855667615, −18.64872207331440299586289706164, −17.49287582400898563874879659067, −16.41546481092218274384172499770, −15.12638979744622095173055583992, −14.48037645171710720645155234357, −12.89832285104490028447207687287, −12.04188782861345695933594327913, −10.331311992459743096063666119961, −9.00298364444570989696851846066, −7.848145804130139298000543981441, −6.96501130219593476323257218068, −6.08365034441934373064742703088, −4.162120057606331596480933815797, −2.17831069021653916025965363073, 0.17229113664798104503556477865, 2.92468801486783791620713710694, 3.76656260371221820507191073841, 4.99542273742914719742987114587, 7.27989366023084731705017449872, 8.66313062030240905367387234355, 9.439226362706609232637024504657, 10.65935407715287851001311192269, 11.53307717715884418872741389529, 12.75804910943596219306870894857, 14.02223807076833523075876334322, 15.69323808674763207475293169418, 16.36598617850100247769901877265, 17.38981321533797622553912934882, 19.13908704393963922864950022459, 19.76006088002051805215970544540, 20.45879113143431056527183524664, 21.880013171205943902523489625614, 22.373318294935824975381181854988, 23.69027341615077905844424138833, 25.37207529465365138355909903359, 26.4861094858538356214558607381, 27.115583131270073512553771430841, 27.84549664589940583623284528582, 28.97974686324735941526821624378

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