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Properties

Label 1-693-693.257-r0-0-0
Degree $1$
Conductor $693$
Sign $0.960 + 0.277i$
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  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.669 + 0.743i)5-s + (0.809 + 0.587i)8-s + (0.5 − 0.866i)10-s + (−0.669 + 0.743i)13-s + (0.309 − 0.951i)16-s + (0.669 + 0.743i)17-s + (0.104 − 0.994i)19-s + (−0.978 − 0.207i)20-s + (0.5 − 0.866i)23-s + (−0.104 + 0.994i)25-s + (0.913 + 0.406i)26-s + (0.104 + 0.994i)29-s + (−0.309 − 0.951i)31-s − 32-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.669 + 0.743i)5-s + (0.809 + 0.587i)8-s + (0.5 − 0.866i)10-s + (−0.669 + 0.743i)13-s + (0.309 − 0.951i)16-s + (0.669 + 0.743i)17-s + (0.104 − 0.994i)19-s + (−0.978 − 0.207i)20-s + (0.5 − 0.866i)23-s + (−0.104 + 0.994i)25-s + (0.913 + 0.406i)26-s + (0.104 + 0.994i)29-s + (−0.309 − 0.951i)31-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.960 + 0.277i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ 0.960 + 0.277i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.144467906 + 0.1618618640i\)
\(L(\frac12)\) \(\approx\) \(1.144467906 + 0.1618618640i\)
\(L(1)\) \(\approx\) \(0.9358804826 - 0.1298640096i\)
\(L(1)\) \(\approx\) \(0.9358804826 - 0.1298640096i\)

Euler product

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

−22.867310097238008066525282428526, −21.93038071621555259253909543305, −21.04571153180146507199380462272, −20.143781159555913439404478202827, −19.25319192810008012103183901024, −18.30576977642932281404075442032, −17.5634561151165127951271229356, −16.904158124165924740509025454406, −16.20880005508245584395681174639, −15.34116012446186218595892327516, −14.39604078668087274700143311741, −13.709093227869456220255625617839, −12.84530021917115274768725335443, −12.00814517823821393661683573359, −10.49860064669261720290788580666, −9.75569949073739233813603503961, −9.12969510271027445933215745022, −8.05895986898758259216334035109, −7.42327041193833264702174545807, −6.21014670550331478062965445055, −5.39960307750896037471498691711, −4.8333269318027315631971564418, −3.46703588510472789825259346707, −1.885871835032430586556627539285, −0.69376083036162914576411234798, 1.26984010189466461720921451998, 2.36394576096938424386183975195, 3.05927496954672570380149379772, 4.25972092552396735089188642332, 5.24129067841727003272697172155, 6.50211064992891151883120263345, 7.401798750271697160058725101463, 8.52942203988347705725123982110, 9.48498975008160904214871420685, 10.07661123976132343738427829203, 10.9824763097897580304181145865, 11.6488020723999891466866298714, 12.749811362605200696641790943111, 13.39657981518667355004022371368, 14.419341923826345926132113904330, 14.93491620496224512921153591740, 16.61276560934873799839267170344, 17.04574533071744809464682690215, 18.13189288330385808700911088936, 18.569578504629790536536578600976, 19.48468576437305413658860710178, 20.15709562888273911844053989523, 21.4206074346923270838306131677, 21.56431624082618333492858984697, 22.45123799461275755512403972009

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