Properties

Label 1-105-105.44-r1-0-0
Degree $1$
Conductor $105$
Sign $-0.895 + 0.444i$
Analytic cond. $11.2838$
Root an. cond. $11.2838$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(11.2838\)
Root analytic conductor: \(11.2838\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 105,\ (1:\ ),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06533879300 - 0.2789141793i\)
\(L(\frac12)\) \(\approx\) \(-0.06533879300 - 0.2789141793i\)
\(L(1)\) \(\approx\) \(0.5493771009 - 0.2723412023i\)
\(L(1)\) \(\approx\) \(0.5493771009 - 0.2723412023i\)

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

−29.89868223375412927209055191513, −28.829048007714046958839570566335, −27.687660240021414207853854844120, −27.0192396528579788513391888038, −25.82261535683761852748668378458, −25.00452398899386431373514057034, −24.084120872907276272492329698855, −22.93090849834523702737738739310, −22.0945345462781017969679419520, −20.38409126647060083294580318779, −19.449403963525748085958778709056, −18.276418771017353473044460801604, −17.32750765027055255176422823665, −16.41407208990373057305120863751, −15.14757933211887186189560277597, −14.4088456420389496956251977844, −13.05527651198810013912438132846, −11.5945825988443723407450962696, −10.02416795740906573723465796983, −9.264072879993976079549714505794, −7.78808413960726810088376298326, −6.887456644001115842130507593974, −5.49838697343202041384613665660, −4.228694793321656657799976381, −1.88921955558837161199913812121, 0.14096964910544635228477804826, 1.93424418653199510222221150194, 3.38397176177659754836981306601, 4.755073300678322130690741390826, 6.64928949692937849352736745075, 8.16382531935513175056389747299, 9.135631861667750098523053841659, 10.38503326779080928226001006594, 11.38762608512067854001826790455, 12.50005525862979035440925920404, 13.56924233613953713961145621733, 14.89930132395357707934589529509, 16.55318097462350856377297785837, 17.308381471025647728030947822775, 18.53881233997781784872207685699, 19.51099632786497457570814517202, 20.27544545598883976714070210571, 21.79244521856759348986315582559, 22.04601782809088400972718170663, 23.686225419326703265294688102343, 24.84344069285568189236667077100, 26.17724397322768005436317674164, 26.896822382989606730414675259218, 27.93179778206207243352133655515, 28.87548302251100609168071667510

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