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Properties

Label 1-297-297.268-r0-0-0
Degree $1$
Conductor $297$
Sign $0.999 + 0.0354i$
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  + (0.0348 − 0.999i)2-s + (−0.997 − 0.0697i)4-s + (−0.882 − 0.469i)5-s + (−0.241 + 0.970i)7-s + (−0.104 + 0.994i)8-s + (−0.5 + 0.866i)10-s + (−0.615 − 0.788i)13-s + (0.961 + 0.275i)14-s + (0.990 + 0.139i)16-s + (0.669 + 0.743i)17-s + (−0.104 + 0.994i)19-s + (0.848 + 0.529i)20-s + (0.766 + 0.642i)23-s + (0.559 + 0.829i)25-s + (−0.809 + 0.587i)26-s + ⋯
L(s)  = 1  + (0.0348 − 0.999i)2-s + (−0.997 − 0.0697i)4-s + (−0.882 − 0.469i)5-s + (−0.241 + 0.970i)7-s + (−0.104 + 0.994i)8-s + (−0.5 + 0.866i)10-s + (−0.615 − 0.788i)13-s + (0.961 + 0.275i)14-s + (0.990 + 0.139i)16-s + (0.669 + 0.743i)17-s + (−0.104 + 0.994i)19-s + (0.848 + 0.529i)20-s + (0.766 + 0.642i)23-s + (0.559 + 0.829i)25-s + (−0.809 + 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.999 + 0.0354i$
Analytic conductor: \(1.37926\)
Root analytic conductor: \(1.37926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (0:\ ),\ 0.999 + 0.0354i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7479620610 + 0.01324384631i\)
\(L(\frac12)\) \(\approx\) \(0.7479620610 + 0.01324384631i\)
\(L(1)\) \(\approx\) \(0.7443362444 - 0.2279822233i\)
\(L(1)\) \(\approx\) \(0.7443362444 - 0.2279822233i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.0348 - 0.999i)T \)
5 \( 1 + (-0.882 - 0.469i)T \)
7 \( 1 + (-0.241 + 0.970i)T \)
13 \( 1 + (-0.615 - 0.788i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.961 - 0.275i)T \)
31 \( 1 + (-0.374 + 0.927i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
41 \( 1 + (0.961 + 0.275i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.997 + 0.0697i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.438 + 0.898i)T \)
61 \( 1 + (-0.374 - 0.927i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (0.0348 - 0.999i)T \)
83 \( 1 + (-0.615 + 0.788i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.882 + 0.469i)T \)
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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

−25.5640120085361084955635507789, −24.228456161037965634263430728541, −23.753990342476348409206837436128, −22.84525296526279630930959973969, −22.273754364165100094617191454990, −20.98245860703656731037826949130, −19.681921144297032052447652170184, −19.036497089840759961930076418809, −18.03922760807063586956948250697, −16.90450576010565115164676357536, −16.336185711987717396633436177446, −15.3311696215281447640001472590, −14.459310825830508264403761258869, −13.707736589172181757491936841084, −12.57498043227574308503171139977, −11.430217830325232957051833718774, −10.261147028103086893602552487903, −9.23176455398060739621377492607, −8.03221382635156148804785851843, −7.10452708774550360982968640951, −6.66412975674172328970550407833, −4.95720796345430480098226252075, −4.17137552622937941194501057109, −3.017098471593200146150874438120, −0.56792444518601671767075195649, 1.268473822121491176647427472317, 2.77231484948169774999109422198, 3.68053190497902183002308369365, 4.91605186486912569594216054104, 5.81138551983959680727410641876, 7.710640995051449788938501025207, 8.50234235446584878453458758251, 9.4747750905747668978434836816, 10.515564533074852001099209983893, 11.60107079208375192434923746109, 12.49745842976142148767407289711, 12.77842804184543438601337782857, 14.396940397878947668090784374771, 15.1776947039788130241949998037, 16.28263930468989700863714597031, 17.42100328861163431605858486733, 18.41602908851716668048342327659, 19.42148315866188758351064806886, 19.712123107631084818187241954035, 21.03656957129755910018446174533, 21.55488878008554730469383833651, 22.8177757914881226810897118851, 23.213926910880777722485455159025, 24.47550385100345992735057420505, 25.3483696267042649418288764985

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