Properties

Label 1-1020-1020.563-r1-0-0
Degree $1$
Conductor $1020$
Sign $-0.184 + 0.982i$
Analytic cond. $109.614$
Root an. cond. $109.614$
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  + (0.707 − 0.707i)7-s + (0.707 + 0.707i)11-s − i·13-s − i·19-s + (−0.707 + 0.707i)23-s + (−0.707 + 0.707i)29-s + (−0.707 + 0.707i)31-s + (−0.707 − 0.707i)37-s + (0.707 + 0.707i)41-s − 43-s − i·47-s − i·49-s + 53-s − i·59-s + (−0.707 − 0.707i)61-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)7-s + (0.707 + 0.707i)11-s − i·13-s − i·19-s + (−0.707 + 0.707i)23-s + (−0.707 + 0.707i)29-s + (−0.707 + 0.707i)31-s + (−0.707 − 0.707i)37-s + (0.707 + 0.707i)41-s − 43-s − i·47-s − i·49-s + 53-s − i·59-s + (−0.707 − 0.707i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.184 + 0.982i$
Analytic conductor: \(109.614\)
Root analytic conductor: \(109.614\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1020} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1020,\ (1:\ ),\ -0.184 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9129818318 + 1.100014885i\)
\(L(\frac12)\) \(\approx\) \(0.9129818318 + 1.100014885i\)
\(L(1)\) \(\approx\) \(1.068159603 + 0.08296845040i\)
\(L(1)\) \(\approx\) \(1.068159603 + 0.08296845040i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 - iT \)
19 \( 1 - iT \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 - T \)
47 \( 1 - iT \)
53 \( 1 + T \)
59 \( 1 - iT \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 + (0.707 + 0.707i)T \)
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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

−21.336862337823521120811525959375, −20.45995032665728714200670324145, −19.57978265921485316524766661841, −18.7655206913071209233323313620, −18.23084142578035641671072746437, −17.17305148493565666415561577597, −16.61417539379755886198380739779, −15.61972953621802616193935104133, −14.86265130051203727478570435097, −14.10370265194151668240062611992, −13.40955831035940831553195844542, −12.24419874796351597234777171393, −11.56026340024187546955663378107, −11.04998737173216940029680267817, −9.80245124216195686209714800476, −8.91992870123804121012711246499, −8.434686662461804853232592665, −7.28431340764528521074022306396, −6.38246833909689136487114091227, −5.55125142274178037325133660446, −4.55764246767468925961853925958, −3.71457846304573983466432150867, −2.431397343374876141476573840720, −1.65521763865474611630777239669, −0.29530251583895886483662420893, 1.17130548182176564367545429132, 1.89408075680774493653264863943, 3.36842861399682068402652571732, 4.09309934069848645042621455456, 5.103250999064449802233534046733, 5.9472911109014871312891412416, 7.1529292728341106131479969690, 7.69294332363462605907333648599, 8.615986633083052426215836996543, 9.67091299724461744126999920319, 10.41100523795486393380731167802, 11.17733859142025294559049888572, 12.13308582556719287070217951878, 12.8190881861378235828010331153, 13.85838303549330934821395150482, 14.53654242582155296050237705911, 15.169015854187639117659980843973, 16.256663496619792603155553969294, 16.975016648522422953969069624691, 17.82210914818176556008249954119, 18.217587713322917719748231840424, 19.564398374044938088275509400581, 20.05810496856668724088453797552, 20.74053744032012248777024948332, 21.57840023009001501040945213278

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