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Properties

Label 1-297-297.184-r1-0-0
Degree $1$
Conductor $297$
Sign $-0.967 + 0.253i$
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  + (0.882 + 0.469i)2-s + (0.559 + 0.829i)4-s + (0.848 + 0.529i)5-s + (−0.961 − 0.275i)7-s + (0.104 + 0.994i)8-s + (0.5 + 0.866i)10-s + (−0.990 − 0.139i)13-s + (−0.719 − 0.694i)14-s + (−0.374 + 0.927i)16-s + (−0.669 + 0.743i)17-s + (0.104 + 0.994i)19-s + (0.0348 + 0.999i)20-s + (−0.939 − 0.342i)23-s + (0.438 + 0.898i)25-s + (−0.809 − 0.587i)26-s + ⋯
L(s)  = 1  + (0.882 + 0.469i)2-s + (0.559 + 0.829i)4-s + (0.848 + 0.529i)5-s + (−0.961 − 0.275i)7-s + (0.104 + 0.994i)8-s + (0.5 + 0.866i)10-s + (−0.990 − 0.139i)13-s + (−0.719 − 0.694i)14-s + (−0.374 + 0.927i)16-s + (−0.669 + 0.743i)17-s + (0.104 + 0.994i)19-s + (0.0348 + 0.999i)20-s + (−0.939 − 0.342i)23-s + (0.438 + 0.898i)25-s + (−0.809 − 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.967 + 0.253i$
Analytic conductor: \(31.9170\)
Root analytic conductor: \(31.9170\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (1:\ ),\ -0.967 + 0.253i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2887372732 + 2.241114716i\)
\(L(\frac12)\) \(\approx\) \(0.2887372732 + 2.241114716i\)
\(L(1)\) \(\approx\) \(1.295530070 + 0.8816973386i\)
\(L(1)\) \(\approx\) \(1.295530070 + 0.8816973386i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.882 + 0.469i)T \)
5 \( 1 + (0.848 + 0.529i)T \)
7 \( 1 + (-0.961 - 0.275i)T \)
13 \( 1 + (-0.990 - 0.139i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (0.104 + 0.994i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.719 - 0.694i)T \)
31 \( 1 + (-0.615 + 0.788i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
41 \( 1 + (0.719 + 0.694i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (0.559 - 0.829i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.997 + 0.0697i)T \)
61 \( 1 + (0.615 + 0.788i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (-0.913 + 0.406i)T \)
79 \( 1 + (0.882 + 0.469i)T \)
83 \( 1 + (-0.990 + 0.139i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.848 - 0.529i)T \)
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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

−24.61422782141569888056041844503, −23.93595195919780724811407103442, −22.71404899603628096233339548933, −21.9486487276475204931923990211, −21.49287050825762990929772147871, −20.13577654892218342004045278879, −19.79837922671697204885640608156, −18.56824468697591477714922721074, −17.48481081955280857973493331223, −16.27852342561467753277537421211, −15.60160894359071240697279077391, −14.33995958568868486769617799830, −13.554250747645256351749250242847, −12.74490214231996838898376241562, −12.030557580760341639620556719960, −10.78087445574344413809324475123, −9.643495559691612252197936391265, −9.189343664958330278885722931190, −7.21193730980782732627910374682, −6.239773527847569982307251006, −5.30981246291075873880365091872, −4.35758002185750389314348461873, −2.87860021919032062766269037295, −2.07796192680104186586882596866, −0.43949355758797082755394175487, 2.03442287344551912849063161283, 3.05846474170800958668741860544, 4.16173290254479718626337476957, 5.5132477215650195397344433231, 6.36881144403277588575917807744, 7.084569385374333221225153353729, 8.357290917978431898526430232098, 9.822016632692643356183445891245, 10.53027023104673556312115181456, 11.97243355500068908981304162514, 12.83064676865895738848496975943, 13.668072041123748329440344979881, 14.47229255456397496535357972154, 15.34927163957531819090479910891, 16.454648369138341009133414781169, 17.16260055296425257040158682153, 18.13164949908331951711884966173, 19.43952727549168449575733619335, 20.314061672560628144677131081269, 21.48673550744270918001161344049, 22.124261503375478439084810264370, 22.74404176136901734301254350379, 23.734909684415496368627874601093, 24.79795936852717758599668363120, 25.41857489129208662023093952359

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