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Properties

Label 1-13e2-169.62-r0-0-0
Degree $1$
Conductor $169$
Sign $-0.981 + 0.190i$
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  + (0.632 + 0.774i)2-s + (−0.996 + 0.0804i)3-s + (−0.200 + 0.979i)4-s + (−0.120 + 0.992i)5-s + (−0.692 − 0.721i)6-s + (0.845 + 0.534i)7-s + (−0.885 + 0.464i)8-s + (0.987 − 0.160i)9-s + (−0.845 + 0.534i)10-s + (−0.987 − 0.160i)11-s + (0.120 − 0.992i)12-s + (0.120 + 0.992i)14-s + (0.0402 − 0.999i)15-s + (−0.919 − 0.391i)16-s + (−0.845 − 0.534i)17-s + (0.748 + 0.663i)18-s + ⋯
L(s)  = 1  + (0.632 + 0.774i)2-s + (−0.996 + 0.0804i)3-s + (−0.200 + 0.979i)4-s + (−0.120 + 0.992i)5-s + (−0.692 − 0.721i)6-s + (0.845 + 0.534i)7-s + (−0.885 + 0.464i)8-s + (0.987 − 0.160i)9-s + (−0.845 + 0.534i)10-s + (−0.987 − 0.160i)11-s + (0.120 − 0.992i)12-s + (0.120 + 0.992i)14-s + (0.0402 − 0.999i)15-s + (−0.919 − 0.391i)16-s + (−0.845 − 0.534i)17-s + (0.748 + 0.663i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.981 + 0.190i$
Analytic conductor: \(0.784832\)
Root analytic conductor: \(0.784832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ -0.981 + 0.190i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09159538868 + 0.9507398989i\)
\(L(\frac12)\) \(\approx\) \(0.09159538868 + 0.9507398989i\)
\(L(1)\) \(\approx\) \(0.6560834406 + 0.7315589646i\)
\(L(1)\) \(\approx\) \(0.6560834406 + 0.7315589646i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (0.632 + 0.774i)T \)
3 \( 1 + (-0.996 + 0.0804i)T \)
5 \( 1 + (-0.120 + 0.992i)T \)
7 \( 1 + (0.845 + 0.534i)T \)
11 \( 1 + (-0.987 - 0.160i)T \)
17 \( 1 + (-0.845 - 0.534i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.632 - 0.774i)T \)
31 \( 1 + (0.970 - 0.239i)T \)
37 \( 1 + (-0.278 + 0.960i)T \)
41 \( 1 + (0.996 - 0.0804i)T \)
43 \( 1 + (0.278 + 0.960i)T \)
47 \( 1 + (0.748 - 0.663i)T \)
53 \( 1 + (0.885 - 0.464i)T \)
59 \( 1 + (0.919 - 0.391i)T \)
61 \( 1 + (-0.0402 - 0.999i)T \)
67 \( 1 + (0.200 + 0.979i)T \)
71 \( 1 + (-0.428 + 0.903i)T \)
73 \( 1 + (0.354 + 0.935i)T \)
79 \( 1 + (-0.748 + 0.663i)T \)
83 \( 1 + (-0.568 + 0.822i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.799 + 0.600i)T \)
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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

−27.56837411044892019759141307784, −26.52806546412060188124822147197, −24.43808142702726830060255613655, −24.10158411590839677412491503467, −23.330689176737162452375418414599, −22.27124015090580079510298046418, −21.23611741271726966287777917706, −20.57585133902355418537002994148, −19.574271400688380188529865734367, −18.16605554774922804352072553499, −17.50828250364093337195128461079, −16.19690542660281269041720445096, −15.28108536848181670328241288196, −13.74473007444799644791985377467, −12.90795167244638711561368583501, −12.07826071788647118957654621193, −11.02966143595401442402973728394, −10.34851437866055443194815170717, −8.87179487608695519707667155188, −7.35468013645064727071968322123, −5.79322864395711346646523570978, −4.85329306747236160082986642251, −4.21500030203052623530601685204, −2.06711255473349752114385952871, −0.728824942448770179653919488953, 2.45968629641863625020836639439, 4.0419501769880888837509640360, 5.2667077406904904442644545966, 6.03093735020308279435963758948, 7.23764649764656048457777201292, 8.103515290675278852159804909694, 9.90150669016957212591992884541, 11.287033250083007096224794601966, 11.78983591097753328101192358509, 13.16198608772718257029348213047, 14.22935570383639183753524997520, 15.42266430740096351584522855388, 15.837529569598991022090692994153, 17.27871002698781504088721145133, 18.05246193891600062071828306738, 18.681265103525049907056672414829, 20.8018580933685180073799778279, 21.59139600658462661026531368039, 22.44085043869068927599130350751, 23.14410404131801092930640750810, 24.07874179119326617788657539892, 24.848072681004254223031350601800, 26.22151864040150195553912026237, 26.90426639155554519330271433160, 27.84743518906586825278651238847

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