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Properties

Label 1-693-693.391-r0-0-0
Degree $1$
Conductor $693$
Sign $0.908 - 0.418i$
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  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (0.104 − 0.994i)5-s + (−0.309 − 0.951i)8-s + 10-s + (0.913 + 0.406i)13-s + (0.913 − 0.406i)16-s + (−0.809 − 0.587i)17-s + (0.309 + 0.951i)19-s + (0.104 + 0.994i)20-s + (−0.5 − 0.866i)23-s + (−0.978 − 0.207i)25-s + (−0.309 + 0.951i)26-s + (−0.669 − 0.743i)29-s + (−0.913 − 0.406i)31-s + (0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (0.104 − 0.994i)5-s + (−0.309 − 0.951i)8-s + 10-s + (0.913 + 0.406i)13-s + (0.913 − 0.406i)16-s + (−0.809 − 0.587i)17-s + (0.309 + 0.951i)19-s + (0.104 + 0.994i)20-s + (−0.5 − 0.866i)23-s + (−0.978 − 0.207i)25-s + (−0.309 + 0.951i)26-s + (−0.669 − 0.743i)29-s + (−0.913 − 0.406i)31-s + (0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.908 - 0.418i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ 0.908 - 0.418i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.083552429 - 0.2376428423i\)
\(L(\frac12)\) \(\approx\) \(1.083552429 - 0.2376428423i\)
\(L(1)\) \(\approx\) \(0.9585804801 + 0.1737218602i\)
\(L(1)\) \(\approx\) \(0.9585804801 + 0.1737218602i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.104 + 0.994i)T \)
5 \( 1 + (0.104 - 0.994i)T \)
13 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.669 - 0.743i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.978 + 0.207i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.978 - 0.207i)T \)
61 \( 1 + (0.913 - 0.406i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.104 + 0.994i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 - T \)
97 \( 1 + (0.104 + 0.994i)T \)
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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

−22.30192360656314714964626393715, −22.081251816160567762786174828168, −21.17681468112283434531677851548, −20.19978960306602639599539008698, −19.57835019738186590968780385640, −18.70982842796235026952274526640, −17.924398916699451413101905874923, −17.53699998423087402242880769216, −16.03953288380468567057206806503, −15.09749532674597129423739245387, −14.3562480331136180811586758998, −13.388764121347985408805976432392, −12.91953899644304667789001542839, −11.52617080250032270239656952948, −11.11669065281782120811284055585, −10.33472891113043640278928890951, −9.4154052368399954630586842850, −8.52828577479757609826011297558, −7.41941392616699336353303973195, −6.27317788400347256118140954869, −5.38806625654898223188527959151, −4.10740462775854631319543704372, −3.32170435478827004499299507403, −2.42349272336398165615854284414, −1.32120815240327127693277568415, 0.55995768445746074704506512848, 2.00995095028354716518550142042, 3.81011364167148527023382431165, 4.376282622000601889166344751, 5.53974604541123973854843350738, 6.10469359176281008486620535519, 7.29017585404362737544388596224, 8.14982181952350786782476427484, 8.99036856392971212945816089042, 9.53160881730138300718159149768, 10.83946342291921483649230072653, 12.069631577155477223747162968945, 12.818964707163825067410879272321, 13.65238392219101716110818884347, 14.27215089063326338212666234359, 15.433066770932970346846690319211, 16.16013311417892397689276305552, 16.6310770727199824209468649290, 17.58448356679189140366645228645, 18.33490515831698014569558260884, 19.155886040312792851223527993510, 20.45644708522810833198447919745, 20.88906260603639187848485878202, 22.04902727383879485279146031783, 22.71804317397440559609982213554

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