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Properties

Label 1-693-693.464-r0-0-0
Degree $1$
Conductor $693$
Sign $0.315 - 0.949i$
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  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.104 + 0.994i)5-s + (0.309 − 0.951i)8-s + (0.5 − 0.866i)10-s + (0.104 − 0.994i)13-s + (−0.809 + 0.587i)16-s + (−0.104 − 0.994i)17-s + (0.978 − 0.207i)19-s + (−0.913 + 0.406i)20-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (−0.669 + 0.743i)26-s + (−0.978 − 0.207i)29-s + (−0.809 − 0.587i)31-s + 32-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.104 + 0.994i)5-s + (0.309 − 0.951i)8-s + (0.5 − 0.866i)10-s + (0.104 − 0.994i)13-s + (−0.809 + 0.587i)16-s + (−0.104 − 0.994i)17-s + (0.978 − 0.207i)19-s + (−0.913 + 0.406i)20-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (−0.669 + 0.743i)26-s + (−0.978 − 0.207i)29-s + (−0.809 − 0.587i)31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.315 - 0.949i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (464, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ 0.315 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6862824755 - 0.4952225595i\)
\(L(\frac12)\) \(\approx\) \(0.6862824755 - 0.4952225595i\)
\(L(1)\) \(\approx\) \(0.7215248499 - 0.1766789965i\)
\(L(1)\) \(\approx\) \(0.7215248499 - 0.1766789965i\)

Euler product

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

−23.211889203407771726325523982740, −21.93383012702778152237986447618, −21.05449749095393411654122330244, −20.19557867238443356415944061757, −19.55054945597418372018078828235, −18.68291686126119183575072581359, −17.83454078667961767164549457632, −16.93158960372266192426752147980, −16.50201414891300701188346094330, −15.6546968427084262123165220593, −14.77121814399340706628691168134, −13.824677835872573370616254961417, −12.97304104811814926828890622318, −11.81744207247583009924641452286, −11.07083135035955808356073701483, −9.854362186101377699893196928164, −9.25649642222302170956813779307, −8.484457757275438623204772607751, −7.62821072854899106248623858552, −6.654262115299374075682988169053, −5.63891088579402657551213108840, −4.91694848001531152255731301751, −3.68701297419195515737739177880, −1.923460563388934859616729480055, −1.20332729974377395720686915228, 0.59421653926531769454790982073, 2.114423749858844455514666522855, 2.953040207471424641652280765605, 3.72101183941107580519101365695, 5.20877908804897475497564246097, 6.452043334924319950535332130066, 7.36511606355373730079374801730, 7.963641323482761665943557685632, 9.26775201011779493036026095813, 9.823590441530320772992919116621, 10.92517927309418743012989170110, 11.23083251651669286316972848805, 12.37941882192005969079320256410, 13.24503189980791966702462714151, 14.20194840252739088145472614434, 15.25603360455191111229093054870, 16.00519902580091074737435034108, 17.01783805130520968374453009148, 17.878080592367796863077644189228, 18.46005558347756983056387071335, 19.03306563642262012978647595524, 20.266202330540630570919762992351, 20.50222058493237121734141704426, 21.75250718898199098615767562374, 22.39020208005177066920156712998

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