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Properties

Label 1-13e2-169.36-r0-0-0
Degree $1$
Conductor $169$
Sign $0.999 - 0.0371i$
Analytic cond. $0.784832$
Root an. cond. $0.784832$
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  + (−0.799 − 0.600i)2-s + (−0.200 − 0.979i)3-s + (0.278 + 0.960i)4-s + (−0.885 + 0.464i)5-s + (−0.428 + 0.903i)6-s + (−0.987 + 0.160i)7-s + (0.354 − 0.935i)8-s + (−0.919 + 0.391i)9-s + (0.987 + 0.160i)10-s + (0.919 + 0.391i)11-s + (0.885 − 0.464i)12-s + (0.885 + 0.464i)14-s + (0.632 + 0.774i)15-s + (−0.845 + 0.534i)16-s + (0.987 − 0.160i)17-s + (0.970 + 0.239i)18-s + ⋯
L(s)  = 1  + (−0.799 − 0.600i)2-s + (−0.200 − 0.979i)3-s + (0.278 + 0.960i)4-s + (−0.885 + 0.464i)5-s + (−0.428 + 0.903i)6-s + (−0.987 + 0.160i)7-s + (0.354 − 0.935i)8-s + (−0.919 + 0.391i)9-s + (0.987 + 0.160i)10-s + (0.919 + 0.391i)11-s + (0.885 − 0.464i)12-s + (0.885 + 0.464i)14-s + (0.632 + 0.774i)15-s + (−0.845 + 0.534i)16-s + (0.987 − 0.160i)17-s + (0.970 + 0.239i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.999 - 0.0371i$
Analytic conductor: \(0.784832\)
Root analytic conductor: \(0.784832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ 0.999 - 0.0371i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4869019010 + 0.009052210980i\)
\(L(\frac12)\) \(\approx\) \(0.4869019010 + 0.009052210980i\)
\(L(1)\) \(\approx\) \(0.5347456169 - 0.1398400722i\)
\(L(1)\) \(\approx\) \(0.5347456169 - 0.1398400722i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (-0.799 - 0.600i)T \)
3 \( 1 + (-0.200 - 0.979i)T \)
5 \( 1 + (-0.885 + 0.464i)T \)
7 \( 1 + (-0.987 + 0.160i)T \)
11 \( 1 + (0.919 + 0.391i)T \)
17 \( 1 + (0.987 - 0.160i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.799 + 0.600i)T \)
31 \( 1 + (-0.568 - 0.822i)T \)
37 \( 1 + (0.996 - 0.0804i)T \)
41 \( 1 + (0.200 + 0.979i)T \)
43 \( 1 + (-0.996 - 0.0804i)T \)
47 \( 1 + (0.970 - 0.239i)T \)
53 \( 1 + (-0.354 + 0.935i)T \)
59 \( 1 + (0.845 + 0.534i)T \)
61 \( 1 + (-0.632 + 0.774i)T \)
67 \( 1 + (-0.278 + 0.960i)T \)
71 \( 1 + (-0.948 - 0.316i)T \)
73 \( 1 + (-0.120 - 0.992i)T \)
79 \( 1 + (-0.970 + 0.239i)T \)
83 \( 1 + (0.748 + 0.663i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.0402 + 0.999i)T \)
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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

−27.39389912709401374804313123375, −26.799316051903993452614258907225, −25.89149822426336987197195434307, −24.902823831732282624329048776766, −23.694280804142106517474981184822, −22.93811317127475489254851915411, −21.932961732304286510521251953594, −20.406262392596493344036586853859, −19.75570023817371481114686516687, −18.92947778835837539990543504246, −17.40583499564704284570736645286, −16.43645774872560376942008383691, −16.1006457159453638782654829090, −15.08070009202089054978753892518, −14.03647624190920059743239948903, −12.222656079104369512037354920242, −11.21094148246086637232166418557, −10.07420985289193762062364158790, −9.19575239110398549379003677592, −8.32211873438668768265793799283, −6.91062154817728712639131279207, −5.789892439955279057177697665433, −4.48015244703438743401310862282, −3.245643302132250179061150695296, −0.63566826625865012150865761369, 1.21017493779875762442862803442, 2.81333556149016038084787882608, 3.79674918248508708825864561570, 6.124016385201293121776828793175, 7.19442986014307283244370815458, 7.90316007735645916769554729703, 9.23569370852102237079443771754, 10.36412883344700549190579724381, 11.84429746914725396322518527072, 12.00791826539504371878950804539, 13.219141657863094889157397769608, 14.608765779117537009413371095459, 16.11941790587123680503531999074, 16.85540921526391666846287526219, 18.160734909509002522980555505944, 18.79931602441687950581938151033, 19.639233862165239689440745183299, 20.169604284275854459890884138767, 21.92983396351456268489998052838, 22.71617310339698894969814385710, 23.58282024336452481130659709518, 25.12421876510773321518136646992, 25.53825202933864576986178731145, 26.75477863646401377358821982483, 27.721635051943691353562249584613

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