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Properties

Label 1-297-297.203-r1-0-0
Degree $1$
Conductor $297$
Sign $0.773 - 0.633i$
Analytic cond. $31.9170$
Root an. cond. $31.9170$
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  + (−0.559 + 0.829i)2-s + (−0.374 − 0.927i)4-s + (−0.438 + 0.898i)5-s + (0.848 − 0.529i)7-s + (0.978 + 0.207i)8-s + (−0.5 − 0.866i)10-s + (0.961 − 0.275i)13-s + (−0.0348 + 0.999i)14-s + (−0.719 + 0.694i)16-s + (0.104 − 0.994i)17-s + (−0.978 − 0.207i)19-s + (0.997 + 0.0697i)20-s + (−0.766 + 0.642i)23-s + (−0.615 − 0.788i)25-s + (−0.309 + 0.951i)26-s + ⋯
L(s)  = 1  + (−0.559 + 0.829i)2-s + (−0.374 − 0.927i)4-s + (−0.438 + 0.898i)5-s + (0.848 − 0.529i)7-s + (0.978 + 0.207i)8-s + (−0.5 − 0.866i)10-s + (0.961 − 0.275i)13-s + (−0.0348 + 0.999i)14-s + (−0.719 + 0.694i)16-s + (0.104 − 0.994i)17-s + (−0.978 − 0.207i)19-s + (0.997 + 0.0697i)20-s + (−0.766 + 0.642i)23-s + (−0.615 − 0.788i)25-s + (−0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.773 - 0.633i$
Analytic conductor: \(31.9170\)
Root analytic conductor: \(31.9170\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (1:\ ),\ 0.773 - 0.633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8685262154 - 0.3100243013i\)
\(L(\frac12)\) \(\approx\) \(0.8685262154 - 0.3100243013i\)
\(L(1)\) \(\approx\) \(0.7344978059 + 0.1788282735i\)
\(L(1)\) \(\approx\) \(0.7344978059 + 0.1788282735i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.559 + 0.829i)T \)
5 \( 1 + (-0.438 + 0.898i)T \)
7 \( 1 + (0.848 - 0.529i)T \)
13 \( 1 + (0.961 - 0.275i)T \)
17 \( 1 + (0.104 - 0.994i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (-0.0348 - 0.999i)T \)
31 \( 1 + (-0.241 + 0.970i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (-0.0348 + 0.999i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (0.374 - 0.927i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.990 - 0.139i)T \)
61 \( 1 + (-0.241 - 0.970i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (0.669 + 0.743i)T \)
79 \( 1 + (0.559 - 0.829i)T \)
83 \( 1 + (-0.961 - 0.275i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.438 + 0.898i)T \)
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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

−25.48311287258394944094143505645, −24.313353890395003481563304808239, −23.592753635267992879293088470254, −22.36643064075860680016261746201, −21.230988435909685975546899346333, −20.87781574263991220511687299409, −19.88797439981067792480241503779, −18.98696396198545145581907446775, −18.1612155289992380370243966217, −17.17949052407661104344618148893, −16.40335353933075491894487277260, −15.32789052384358286287596189163, −14.05143155538155660944206914275, −12.82770199395766764894051205114, −12.237813643381497395420570015065, −11.2493400657233824860316761651, −10.45839279363047414757132422487, −8.94822207278790228429979084740, −8.55103602563844580807651477582, −7.65534088645014019163226002066, −5.916651904861496828272822474067, −4.53727605914331392137073308378, −3.78691518802257287835492891900, −2.10498700518377278987044937678, −1.198217358118475291983621854272, 0.3718295517345317195275281983, 1.88290815810319711777130297955, 3.65224727074123959775649223957, 4.81394721385819816737725944038, 6.08299468264534800185556317822, 7.06836420497224095602088373254, 7.863190299082027775490240484439, 8.72780308517813968883206670120, 10.13201082108743695190178973288, 10.84006290737336484889154382790, 11.71818179718626585818002481020, 13.56589478412182742936818178405, 14.142841840661703910422372350088, 15.17025214786306116187774898407, 15.80015374251386095317781069023, 16.94669175487643050432736711396, 17.90593250948934313764170503598, 18.44239416595593472500568286705, 19.458570647964521389296385222511, 20.36256625030220452428783734570, 21.55608272165343311261069438089, 22.8743183985649847189449682913, 23.31345045492782374420888135526, 24.16717201087408645528619694629, 25.25504896220781094835938848874

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