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Properties

Label 1-675-675.317-r0-0-0
Degree $1$
Conductor $675$
Sign $-0.722 + 0.691i$
Analytic cond. $3.13468$
Root an. cond. $3.13468$
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  + (0.970 + 0.241i)2-s + (0.882 + 0.469i)4-s + (−0.984 + 0.173i)7-s + (0.743 + 0.669i)8-s + (−0.961 + 0.275i)11-s + (−0.970 + 0.241i)13-s + (−0.997 − 0.0697i)14-s + (0.559 + 0.829i)16-s + (−0.207 + 0.978i)17-s + (−0.669 + 0.743i)19-s + (−0.999 + 0.0348i)22-s + (−0.898 + 0.438i)23-s − 26-s + (−0.951 − 0.309i)28-s + (−0.374 − 0.927i)29-s + ⋯
L(s)  = 1  + (0.970 + 0.241i)2-s + (0.882 + 0.469i)4-s + (−0.984 + 0.173i)7-s + (0.743 + 0.669i)8-s + (−0.961 + 0.275i)11-s + (−0.970 + 0.241i)13-s + (−0.997 − 0.0697i)14-s + (0.559 + 0.829i)16-s + (−0.207 + 0.978i)17-s + (−0.669 + 0.743i)19-s + (−0.999 + 0.0348i)22-s + (−0.898 + 0.438i)23-s − 26-s + (−0.951 − 0.309i)28-s + (−0.374 − 0.927i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.722 + 0.691i$
Analytic conductor: \(3.13468\)
Root analytic conductor: \(3.13468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (0:\ ),\ -0.722 + 0.691i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5464702989 + 1.361676130i\)
\(L(\frac12)\) \(\approx\) \(0.5464702989 + 1.361676130i\)
\(L(1)\) \(\approx\) \(1.251395382 + 0.5711708183i\)
\(L(1)\) \(\approx\) \(1.251395382 + 0.5711708183i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.970 + 0.241i)T \)
7 \( 1 + (-0.984 + 0.173i)T \)
11 \( 1 + (-0.961 + 0.275i)T \)
13 \( 1 + (-0.970 + 0.241i)T \)
17 \( 1 + (-0.207 + 0.978i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 + (-0.898 + 0.438i)T \)
29 \( 1 + (-0.374 - 0.927i)T \)
31 \( 1 + (0.990 - 0.139i)T \)
37 \( 1 + (0.406 + 0.913i)T \)
41 \( 1 + (0.241 + 0.970i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (0.139 - 0.990i)T \)
53 \( 1 + (0.951 + 0.309i)T \)
59 \( 1 + (0.961 + 0.275i)T \)
61 \( 1 + (-0.719 - 0.694i)T \)
67 \( 1 + (-0.927 - 0.374i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (0.406 - 0.913i)T \)
79 \( 1 + (0.374 + 0.927i)T \)
83 \( 1 + (-0.529 + 0.848i)T \)
89 \( 1 + (0.913 + 0.406i)T \)
97 \( 1 + (-0.788 - 0.615i)T \)
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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

−22.383594546879477502881662132188, −21.83418103741428524407407111974, −20.86754674970531232841333065964, −20.07889006059141303349524208655, −19.41924846337659472768866794817, −18.599792929017065771990285614881, −17.44108534030414698642343818291, −16.26444614330283385339287190120, −15.8693202539693929986243928700, −14.930613931641112999107819808026, −13.98207117624367109128706744717, −13.21687450001509483098996775695, −12.57994185030767787195882574061, −11.75544144984769800296033110648, −10.61002473918668630640742698111, −10.1025845788057748475762356181, −8.98209835692785271638847822471, −7.532952597689970363434370071795, −6.8804748796235188800809755085, −5.84625680781685076908014771477, −5.00099971146622154992073651150, −4.04241635546561453851593396017, −2.86218680435264850441461634521, −2.35898170635148552874915683294, −0.45324800381059454576507341312, 1.99001563688579297710054512459, 2.790464066922587179117882382050, 3.880825322855144118215560401742, 4.741986061157230711632269281734, 5.88881866677512721187814805685, 6.43826866124459235330689726187, 7.57614396904940138973443647821, 8.29048441053529879739982414515, 9.82023926032497469903406437052, 10.36783937312279993043664804598, 11.65164191481580564477662490731, 12.40796917831689246854845372535, 13.081361826716911846852767990084, 13.782872396361808530692083962373, 15.058664220293135078281078753714, 15.29190760676028086505009339029, 16.452453397013048407042227653000, 16.94186011893772698626871224729, 18.10409854390046264257921600890, 19.29405269546056591308666076400, 19.7774614399894180686634568358, 20.87877808065230810996569485417, 21.56642623093786120015296552717, 22.28198363602482961094526319634, 23.08218708389544559567858673197

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