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Properties

Label 1-13e2-169.112-r1-0-0
Degree $1$
Conductor $169$
Sign $-0.986 + 0.166i$
Analytic cond. $18.1615$
Root an. cond. $18.1615$
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  + (−0.239 − 0.970i)2-s + (0.120 + 0.992i)3-s + (−0.885 + 0.464i)4-s + (−0.822 + 0.568i)5-s + (0.935 − 0.354i)6-s + (0.663 + 0.748i)7-s + (0.663 + 0.748i)8-s + (−0.970 + 0.239i)9-s + (0.748 + 0.663i)10-s + (−0.239 + 0.970i)11-s + (−0.568 − 0.822i)12-s + (0.568 − 0.822i)14-s + (−0.663 − 0.748i)15-s + (0.568 − 0.822i)16-s + (0.748 − 0.663i)17-s + (0.464 + 0.885i)18-s + ⋯
L(s)  = 1  + (−0.239 − 0.970i)2-s + (0.120 + 0.992i)3-s + (−0.885 + 0.464i)4-s + (−0.822 + 0.568i)5-s + (0.935 − 0.354i)6-s + (0.663 + 0.748i)7-s + (0.663 + 0.748i)8-s + (−0.970 + 0.239i)9-s + (0.748 + 0.663i)10-s + (−0.239 + 0.970i)11-s + (−0.568 − 0.822i)12-s + (0.568 − 0.822i)14-s + (−0.663 − 0.748i)15-s + (0.568 − 0.822i)16-s + (0.748 − 0.663i)17-s + (0.464 + 0.885i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.986 + 0.166i$
Analytic conductor: \(18.1615\)
Root analytic conductor: \(18.1615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (1:\ ),\ -0.986 + 0.166i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04594777366 + 0.5479918788i\)
\(L(\frac12)\) \(\approx\) \(0.04594777366 + 0.5479918788i\)
\(L(1)\) \(\approx\) \(0.6733605047 + 0.1771152558i\)
\(L(1)\) \(\approx\) \(0.6733605047 + 0.1771152558i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (-0.239 - 0.970i)T \)
3 \( 1 + (0.120 + 0.992i)T \)
5 \( 1 + (-0.822 + 0.568i)T \)
7 \( 1 + (0.663 + 0.748i)T \)
11 \( 1 + (-0.239 + 0.970i)T \)
17 \( 1 + (0.748 - 0.663i)T \)
19 \( 1 + iT \)
23 \( 1 - T \)
29 \( 1 + (-0.970 + 0.239i)T \)
31 \( 1 + (0.935 - 0.354i)T \)
37 \( 1 + (-0.935 + 0.354i)T \)
41 \( 1 + (-0.992 + 0.120i)T \)
43 \( 1 + (0.354 - 0.935i)T \)
47 \( 1 + (0.464 - 0.885i)T \)
53 \( 1 + (-0.748 + 0.663i)T \)
59 \( 1 + (0.822 - 0.568i)T \)
61 \( 1 + (-0.748 - 0.663i)T \)
67 \( 1 + (-0.464 + 0.885i)T \)
71 \( 1 + (-0.992 + 0.120i)T \)
73 \( 1 + (-0.239 + 0.970i)T \)
79 \( 1 + (0.885 + 0.464i)T \)
83 \( 1 + (0.992 + 0.120i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.822 + 0.568i)T \)
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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

−26.67829145911064513509432769341, −25.95207009204890768535114434153, −24.64878781567781248731482877357, −23.90710060362545389445569208496, −23.72027286636322482842022125001, −22.51454045223975149097026147801, −20.8885832615063252797459740142, −19.62718164989377463894047719659, −19.06602369734039593562403913868, −17.8773521464390009668998170917, −17.044803207848797995288978792339, −16.16549574086758046176245254844, −14.933432818281876808562410774180, −13.90881945323232539679150636206, −13.15516921935206144033655431512, −11.8891412761594395765749695819, −10.68943086035959134734847206068, −8.92325195893680034309315372323, −8.03648737057332117430581481437, −7.51729647387518050966365945138, −6.21490465686219242551033485018, −4.99750251498269944700972461035, −3.63898873371102749651421520473, −1.295672878466419405227997028396, −0.234829689387682521632211923440, 2.1016243154378401789412183265, 3.31015477269689941295300633100, 4.33894915031262122518172172340, 5.40423443716861386036881653828, 7.656330885866746320225588493415, 8.50920052262728153810277257219, 9.7747379179989580553894444454, 10.52298058069970135286753401107, 11.71161907642080758502700604281, 12.16688856556210158795105764311, 14.0423659653361047076743748917, 14.862410605350086651435241605369, 15.816405115532279151683997489953, 17.11019086658024500585604281169, 18.29061663325084166282603431291, 19.01295214677082418393141359853, 20.37172225416981437362381259694, 20.73152686082791284864758556232, 21.966886310092001265848012280033, 22.602936549235873798661958017291, 23.51594284481043406392008038683, 25.294207856116475777437372560958, 26.23428972374510393049165160830, 27.13390800170056019717086222529, 27.83395023316847636565962787516

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