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Properties

Label 1-105-105.59-r0-0-0
Degree $1$
Conductor $105$
Sign $0.386 + 0.922i$
Analytic cond. $0.487617$
Root an. cond. $0.487617$
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 + (−0.5 − 0.866i)4-s + 8-s + (0.5 + 0.866i)11-s + 13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s − 22-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)26-s − 29-s + (0.5 + 0.866i)31-s + (−0.5 − 0.866i)32-s − 34-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + 8-s + (0.5 + 0.866i)11-s + 13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s − 22-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)26-s − 29-s + (0.5 + 0.866i)31-s + (−0.5 − 0.866i)32-s − 34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(0.487617\)
Root analytic conductor: \(0.487617\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 105,\ (0:\ ),\ 0.386 + 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6730029537 + 0.4476693767i\)
\(L(\frac12)\) \(\approx\) \(0.6730029537 + 0.4476693767i\)
\(L(1)\) \(\approx\) \(0.7721772126 + 0.3354575615i\)
\(L(1)\) \(\approx\) \(0.7721772126 + 0.3354575615i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + T \)
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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

−29.55063620363174332384967920088, −28.49398773687208077611995867342, −27.55471076493807402744282641171, −26.69408880793993815399864927130, −25.64825249355687005709866486761, −24.508933732720135495690334257880, −22.993134384342081884268641195381, −22.14245979176879200587851869259, −20.93977522450069279830948365152, −20.25265319228735948275503312766, −18.85682074172794978410553092368, −18.35714479018807150888995580508, −16.8947099596961447093586393365, −16.10961160651346592965979004347, −14.251734502342626427741275041244, −13.30297707968846412090328293911, −11.9689430358666968784080097425, −11.12512680256273098534849725128, −9.895018542175796583937380594954, −8.77882383984514178647794039729, −7.70295176931696467343010636395, −5.96361201895816372412625415503, −4.15084838673797721076819382901, −2.96816801744910353619330370917, −1.203841995365613031602871983954, 1.54027513790080690107604864956, 3.93055921992593561892928020278, 5.41286467610729772623232911286, 6.61203603088246892937533472389, 7.753044897707870125377564437074, 8.97168942387566330752542302354, 9.99811470607291803293985850613, 11.30569508208003692406688093935, 12.93853382706588541244033355582, 14.126834020169100085955329601376, 15.186414487778433778920498809316, 16.11567857513099670067088766968, 17.315900666341649601686591997544, 18.084420879505044494816759214116, 19.29281543354412835358893719666, 20.25537485576099624565829186243, 21.782879282980770866458178160411, 23.0085467390383359585825347466, 23.75597324284842312259179718748, 24.96184594939417439920108290418, 25.774799624498826348339933602424, 26.62386461917235478766880183415, 28.0671940549055669583507069169, 28.24126854766429845898833027888, 29.944453750688843336748295476856

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