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Properties

Label 1-693-693.571-r1-0-0
Degree $1$
Conductor $693$
Sign $-0.592 + 0.805i$
Analytic cond. $74.4731$
Root an. cond. $74.4731$
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 + 5-s − 8-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + 23-s + 25-s + (−0.5 + 0.866i)26-s + (0.5 − 0.866i)29-s + (−0.5 + 0.866i)31-s + (0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + 5-s − 8-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + 23-s + 25-s + (−0.5 + 0.866i)26-s + (0.5 − 0.866i)29-s + (−0.5 + 0.866i)31-s + (0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.592 + 0.805i$
Analytic conductor: \(74.4731\)
Root analytic conductor: \(74.4731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (1:\ ),\ -0.592 + 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.494771297 + 2.957147692i\)
\(L(\frac12)\) \(\approx\) \(1.494771297 + 2.957147692i\)
\(L(1)\) \(\approx\) \(1.331515804 + 0.9805618352i\)
\(L(1)\) \(\approx\) \(1.331515804 + 0.9805618352i\)

Euler product

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

−22.28187942726248753804446335327, −21.088054107972184374724397538470, −20.84706916669036287472776002696, −19.99568865231041473095412377569, −18.896580238615411130061328326627, −18.23510150669294907434840799790, −17.5727788006614431349538328943, −16.44586237683268794903576330097, −15.42099954458853499959656720470, −14.321490478976943950114383494035, −13.94414605501065982499566654225, −12.87605821556058857551280122649, −12.435005767024436743339194230008, −11.15226368364941416655440253326, −10.561381630704177321016623521813, −9.61196699213698033196952184988, −9.04519395650293896239714971901, −7.708239177164717339792502409431, −6.36735907507657771735512960429, −5.55113415131491800227195686446, −4.90511935052472290788149960672, −3.504917242771221685244941827362, −2.7780014157460956226630157903, −1.62691709063300759730490016314, −0.71146846000462944745295697003, 1.17941194204718264695414217238, 2.54483960091909108840216640769, 3.616665459647481502674658777553, 4.75826013709516611264111459831, 5.53651526724043814737768274650, 6.465025963425608000623396682030, 7.061655173127840237499053555804, 8.35604738917888358099407389606, 9.05400321769381458181441533617, 9.93133813464756331804782587341, 11.115931454645714003172219811649, 12.16263112825669888874211703635, 13.13297181915354785851188621995, 13.67984367763454705867852840987, 14.461538727327951569648738993256, 15.263396254626674914740777184218, 16.25042605912389790024572204596, 16.96262418618433125372397346672, 17.63868833902703803590362259013, 18.42032181232578662874761346343, 19.37425375739560020552811466343, 20.74895409517223487836122143799, 21.339765291259354092787927503867, 21.9131744043714955172714174045, 22.832269780453827655351402281

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