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Properties

Label 1-693-693.2-r0-0-0
Degree $1$
Conductor $693$
Sign $0.0284 - 0.999i$
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  + (0.913 − 0.406i)2-s + (0.669 − 0.743i)4-s + (0.809 − 0.587i)5-s + (0.309 − 0.951i)8-s + (0.5 − 0.866i)10-s + (−0.913 + 0.406i)13-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (0.978 − 0.207i)19-s + (0.104 − 0.994i)20-s − 23-s + (0.309 − 0.951i)25-s + (−0.669 + 0.743i)26-s + (0.669 − 0.743i)29-s + (−0.104 + 0.994i)31-s + (−0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (0.669 − 0.743i)4-s + (0.809 − 0.587i)5-s + (0.309 − 0.951i)8-s + (0.5 − 0.866i)10-s + (−0.913 + 0.406i)13-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (0.978 − 0.207i)19-s + (0.104 − 0.994i)20-s − 23-s + (0.309 − 0.951i)25-s + (−0.669 + 0.743i)26-s + (0.669 − 0.743i)29-s + (−0.104 + 0.994i)31-s + (−0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0284 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0284 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.0284 - 0.999i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ 0.0284 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.945401078 - 2.001497525i\)
\(L(\frac12)\) \(\approx\) \(1.945401078 - 2.001497525i\)
\(L(1)\) \(\approx\) \(1.755431366 - 0.8993477296i\)
\(L(1)\) \(\approx\) \(1.755431366 - 0.8993477296i\)

Euler product

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

−22.62896994017250907821275074966, −22.13728198871588485858921536010, −21.603141773071662377389353316121, −20.6005708602188434361472542625, −19.88491397103907394037038605847, −18.75137785996426586657232122553, −17.66026193484169597018650505403, −17.22307273412318864218630387776, −16.17505154804205401591995303463, −15.300616166696672341145787691, −14.46161211685656086797641712072, −13.98962174929045435333109878918, −12.98844604741071478553032812853, −12.30832887500573620333203857478, −11.27636686779826367119583391428, −10.364394129688783635085940138580, −9.514730557325541450610788654590, −8.13773232889560861105106985291, −7.37016613685316343872853960691, −6.37033436811971898687175802613, −5.71722774001921673015391438465, −4.79590127759373748136198368089, −3.622313607951668133814519731392, −2.6994547587618937896701614599, −1.77935142976389284685606320540, 1.00903013169338983824239730204, 2.166098084766720470585147232873, 2.946391969054058309308979763703, 4.367487658180402501955499869187, 5.00105717497118823118536406975, 5.875578702025572066290416760762, 6.793484661902101671576700918348, 7.84486969829697869118537040293, 9.42131786231196453407280340322, 9.702001477977653181928817956469, 10.84677420231362939842320077952, 11.91552749346899374678162150704, 12.39272182259970662918090116527, 13.49679824589931220778273813656, 13.97101344535711488554797530472, 14.750449201514200698636479133368, 15.99823796623540146481544782459, 16.40993203512806436403856576151, 17.66190190356058802554950755371, 18.355500156571043574615097506780, 19.69884041004198742797582983793, 20.01736337716014763552487275157, 21.094003781271258266220073438441, 21.56060629531122692192804825398, 22.3718580047330726534061888132

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