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Properties

Label 1-297-297.124-r0-0-0
Degree $1$
Conductor $297$
Sign $0.320 + 0.947i$
Analytic cond. $1.37926$
Root an. cond. $1.37926$
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  + (0.990 + 0.139i)2-s + (0.961 + 0.275i)4-s + (−0.374 + 0.927i)5-s + (0.559 + 0.829i)7-s + (0.913 + 0.406i)8-s + (−0.5 + 0.866i)10-s + (−0.882 − 0.469i)13-s + (0.438 + 0.898i)14-s + (0.848 + 0.529i)16-s + (−0.978 − 0.207i)17-s + (0.913 + 0.406i)19-s + (−0.615 + 0.788i)20-s + (−0.939 + 0.342i)23-s + (−0.719 − 0.694i)25-s + (−0.809 − 0.587i)26-s + ⋯
L(s)  = 1  + (0.990 + 0.139i)2-s + (0.961 + 0.275i)4-s + (−0.374 + 0.927i)5-s + (0.559 + 0.829i)7-s + (0.913 + 0.406i)8-s + (−0.5 + 0.866i)10-s + (−0.882 − 0.469i)13-s + (0.438 + 0.898i)14-s + (0.848 + 0.529i)16-s + (−0.978 − 0.207i)17-s + (0.913 + 0.406i)19-s + (−0.615 + 0.788i)20-s + (−0.939 + 0.342i)23-s + (−0.719 − 0.694i)25-s + (−0.809 − 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.320 + 0.947i$
Analytic conductor: \(1.37926\)
Root analytic conductor: \(1.37926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (0:\ ),\ 0.320 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.800641718 + 1.291618702i\)
\(L(\frac12)\) \(\approx\) \(1.800641718 + 1.291618702i\)
\(L(1)\) \(\approx\) \(1.694144000 + 0.6380581422i\)
\(L(1)\) \(\approx\) \(1.694144000 + 0.6380581422i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.990 + 0.139i)T \)
5 \( 1 + (-0.374 + 0.927i)T \)
7 \( 1 + (0.559 + 0.829i)T \)
13 \( 1 + (-0.882 - 0.469i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (0.438 - 0.898i)T \)
31 \( 1 + (0.0348 + 0.999i)T \)
37 \( 1 + (0.913 - 0.406i)T \)
41 \( 1 + (0.438 + 0.898i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.961 - 0.275i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.241 - 0.970i)T \)
61 \( 1 + (0.0348 - 0.999i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (0.990 + 0.139i)T \)
83 \( 1 + (-0.882 + 0.469i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.374 - 0.927i)T \)
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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

−24.71807402232376732123523252968, −24.11972106191400555868349012844, −23.70467327452170671393868187447, −22.48263582581793503932309070052, −21.71766562573543317592342910378, −20.5995923980609907883997153908, −20.117915674091439017692262491334, −19.37714801246734289330447996816, −17.76376832497049180085007080664, −16.75824765830153345988296233270, −16.04194222751267453796077527200, −15.00370327064771597251043682679, −14.01334054678914868004984418635, −13.273142059684876357683203821775, −12.23458707034068667435220705355, −11.51261826652725842495605535174, −10.49657149597928022410130589626, −9.24471306559886262685185991391, −7.842942678134959058458421952654, −7.07872682548862942673567448100, −5.68757715107887982602669445241, −4.52482845187939871949186826279, −4.154781823949846642273703600889, −2.4847560563272372061199803112, −1.14966247100462094236253568756, 2.13017803168992016729466504894, 2.9296894070621187976271198521, 4.186310001760543752903279442370, 5.29437258469125370515202753617, 6.2661130458613775914124902716, 7.37698258367919008096047690307, 8.149985390819100194610184556441, 9.80841774261703048399794979620, 10.98889826683631986721512757750, 11.75012868169819734129713940116, 12.4838302562975690720758530866, 13.85163055902287632452101086324, 14.52009298493945639294363018109, 15.400931958165234430192903675028, 15.95080272391547894813145254806, 17.47261144509647935936550082083, 18.254700384200941299988859310217, 19.48102791527431384023622720471, 20.222292250305792287617694861251, 21.4601234638574454136262592507, 22.08862136586879537429303220723, 22.72596902186066527129423092434, 23.747018026375035655569697008945, 24.65238851276116195162031048099, 25.2418910879304957619302939494

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