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Properties

Label 1-693-693.569-r0-0-0
Degree $1$
Conductor $693$
Sign $-0.535 - 0.844i$
Analytic cond. $3.21827$
Root an. cond. $3.21827$
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  + (0.669 + 0.743i)2-s + (−0.104 + 0.994i)4-s + (−0.309 − 0.951i)5-s + (−0.809 + 0.587i)8-s + (0.5 − 0.866i)10-s + (−0.669 − 0.743i)13-s + (−0.978 − 0.207i)16-s + (−0.978 − 0.207i)17-s + (−0.913 − 0.406i)19-s + (0.978 − 0.207i)20-s − 23-s + (−0.809 + 0.587i)25-s + (0.104 − 0.994i)26-s + (−0.104 + 0.994i)29-s + (−0.978 + 0.207i)31-s + (−0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (−0.104 + 0.994i)4-s + (−0.309 − 0.951i)5-s + (−0.809 + 0.587i)8-s + (0.5 − 0.866i)10-s + (−0.669 − 0.743i)13-s + (−0.978 − 0.207i)16-s + (−0.978 − 0.207i)17-s + (−0.913 − 0.406i)19-s + (0.978 − 0.207i)20-s − 23-s + (−0.809 + 0.587i)25-s + (0.104 − 0.994i)26-s + (−0.104 + 0.994i)29-s + (−0.978 + 0.207i)31-s + (−0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.535 - 0.844i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ -0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09655925740 - 0.1755857900i\)
\(L(\frac12)\) \(\approx\) \(0.09655925740 - 0.1755857900i\)
\(L(1)\) \(\approx\) \(0.8992582835 + 0.2383820917i\)
\(L(1)\) \(\approx\) \(0.8992582835 + 0.2383820917i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.669 + 0.743i)T \)
5 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (-0.669 - 0.743i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.104 + 0.994i)T \)
53 \( 1 + (-0.669 - 0.743i)T \)
59 \( 1 + (0.104 - 0.994i)T \)
61 \( 1 + (0.978 + 0.207i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.913 + 0.406i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (0.669 - 0.743i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.978 + 0.207i)T \)
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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

−22.83922304865140056464572937467, −21.94490186334069823062070270057, −21.66095399378068694953667190919, −20.515011942667477483320837657826, −19.65706911731887452785781188113, −19.09480773709789418656224421181, −18.332748294019085688861959641499, −17.432164841734216572640133252876, −16.1636149248716011773925373836, −15.21366300916830498006438657657, −14.611076010003715120690773135654, −13.890342100114491639941459454890, −12.95538723246679506001094732366, −11.9999528467147340014441479584, −11.33702801721653699961728219817, −10.53266749558550946046405935989, −9.79206535446376114513808172681, −8.726718033645279730170789579761, −7.399202167957009727980909975198, −6.51230082736423201730580623984, −5.71722777403229522206360653606, −4.30530656038382170551038370591, −3.87376887226226828645855431253, −2.511713395782418466895977794180, −1.96055008833754395835777258185, 0.068652738066143579847436083009, 2.00920071035255105616581691057, 3.2686263883342809669863185247, 4.36771528697197140258686326609, 4.96676343588055584286103056878, 5.91339782753819542185560705763, 6.956543201099164309328737719, 7.87000628134126951535877635992, 8.62078658187299920367321769963, 9.425529143690048366652599252105, 10.82203973618693270754997153294, 11.84953646701356688544236755197, 12.6934792185706729324006961725, 13.11412760559029424480707420653, 14.19680670072658994121289534283, 15.064055536846725957070816795266, 15.82030733612129948399595527853, 16.45430324200489037168526882075, 17.407047882495712866234762482204, 17.87878443685627091945422889841, 19.30855135550100684032087367274, 20.20954031596990405183397422706, 20.7391905147680696009713723249, 21.93414320688185667230807922670, 22.28188144619040292379599957303

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