↑

Properties

Label 1-13e2-169.131-r0-0-0
Degree $1$
Conductor $169$
Sign $0.683 - 0.729i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.885 − 0.464i)2-s + (−0.970 + 0.239i)3-s + (0.568 − 0.822i)4-s + (−0.354 + 0.935i)5-s + (−0.748 + 0.663i)6-s + (0.120 − 0.992i)7-s + (0.120 − 0.992i)8-s + (0.885 − 0.464i)9-s + (0.120 + 0.992i)10-s + (0.885 + 0.464i)11-s + (−0.354 + 0.935i)12-s + (−0.354 − 0.935i)14-s + (0.120 − 0.992i)15-s + (−0.354 − 0.935i)16-s + (0.120 − 0.992i)17-s + (0.568 − 0.822i)18-s + ⋯
L(s)  = 1  + (0.885 − 0.464i)2-s + (−0.970 + 0.239i)3-s + (0.568 − 0.822i)4-s + (−0.354 + 0.935i)5-s + (−0.748 + 0.663i)6-s + (0.120 − 0.992i)7-s + (0.120 − 0.992i)8-s + (0.885 − 0.464i)9-s + (0.120 + 0.992i)10-s + (0.885 + 0.464i)11-s + (−0.354 + 0.935i)12-s + (−0.354 − 0.935i)14-s + (0.120 − 0.992i)15-s + (−0.354 − 0.935i)16-s + (0.120 − 0.992i)17-s + (0.568 − 0.822i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.683 - 0.729i$
Analytic conductor: \(0.784832\)
Root analytic conductor: \(0.784832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ 0.683 - 0.729i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.295598060 - 0.5615133831i\)
\(L(\frac12)\) \(\approx\) \(1.295598060 - 0.5615133831i\)
\(L(1)\) \(\approx\) \(1.268189624 - 0.3373617378i\)
\(L(1)\) \(\approx\) \(1.268189624 - 0.3373617378i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−27.88651261074955061704865888566, −26.86263565286334990856863233543, −25.18461138025577310361833083616, −24.62855251944673883983547288805, −23.89601262508483715314489663455, −22.99299133145382509472184287806, −21.92970965338582813703144467142, −21.41990688612430209258264368017, −20.080536978200867882666499338231, −18.85655905915013400593939334584, −17.487916193172931598425067505797, −16.725429154954688628364858878695, −15.910109046775552576485075367804, −14.97563873382243107654683936161, −13.55483579313420893643207395309, −12.471834705957748519709948289914, −11.988628592613521960092079828925, −11.05227199500354124199188513247, −9.096020311083055520288477304120, −8.00454428967281893202703812754, −6.66968574300465132629524808503, −5.62163805301369504840836504991, −4.917635047125664308733281570708, −3.596845655541757622639811907544, −1.60070110083289774435875825399, 1.206662595099503854441920780901, 3.190575916457331903307589581916, 4.20611890135551728995030399679, 5.246548712455300933791396628265, 6.79123755035835885647126819325, 7.10829868914295890136482949034, 9.72392738058294178653893188928, 10.514947090249089956835455494622, 11.466545234169376286686742061250, 12.04965118082563983880994410141, 13.52032984192250304775897807885, 14.42834651011740935809030168115, 15.45066181769036749282507417135, 16.452511428393249870979171665183, 17.65293264448484990097870355618, 18.73122203731236945824236301620, 19.8603968550645965826285770775, 20.78911066674607880172938691370, 21.97896420894474844619770758347, 22.71537289190267517327239171978, 23.17330606765561753117191155195, 24.138918772191026467433390370968, 25.3534472813370016912795997317, 27.0188845374965916161785357972, 27.31350296179807313186460152875

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