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Properties

Label 1-13e2-169.140-r0-0-0
Degree $1$
Conductor $169$
Sign $-0.973 - 0.227i$
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  + (0.0402 − 0.999i)2-s + (0.278 − 0.960i)3-s + (−0.996 − 0.0804i)4-s + (0.354 − 0.935i)5-s + (−0.948 − 0.316i)6-s + (0.919 − 0.391i)7-s + (−0.120 + 0.992i)8-s + (−0.845 − 0.534i)9-s + (−0.919 − 0.391i)10-s + (0.845 − 0.534i)11-s + (−0.354 + 0.935i)12-s + (−0.354 − 0.935i)14-s + (−0.799 − 0.600i)15-s + (0.987 + 0.160i)16-s + (−0.919 + 0.391i)17-s + (−0.568 + 0.822i)18-s + ⋯
L(s)  = 1  + (0.0402 − 0.999i)2-s + (0.278 − 0.960i)3-s + (−0.996 − 0.0804i)4-s + (0.354 − 0.935i)5-s + (−0.948 − 0.316i)6-s + (0.919 − 0.391i)7-s + (−0.120 + 0.992i)8-s + (−0.845 − 0.534i)9-s + (−0.919 − 0.391i)10-s + (0.845 − 0.534i)11-s + (−0.354 + 0.935i)12-s + (−0.354 − 0.935i)14-s + (−0.799 − 0.600i)15-s + (0.987 + 0.160i)16-s + (−0.919 + 0.391i)17-s + (−0.568 + 0.822i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.973 - 0.227i$
Analytic conductor: \(0.784832\)
Root analytic conductor: \(0.784832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ -0.973 - 0.227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1450141684 - 1.259472213i\)
\(L(\frac12)\) \(\approx\) \(0.1450141684 - 1.259472213i\)
\(L(1)\) \(\approx\) \(0.6535184445 - 0.9821024849i\)
\(L(1)\) \(\approx\) \(0.6535184445 - 0.9821024849i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (0.0402 - 0.999i)T \)
3 \( 1 + (0.278 - 0.960i)T \)
5 \( 1 + (0.354 - 0.935i)T \)
7 \( 1 + (0.919 - 0.391i)T \)
11 \( 1 + (0.845 - 0.534i)T \)
17 \( 1 + (-0.919 + 0.391i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.0402 + 0.999i)T \)
31 \( 1 + (0.748 - 0.663i)T \)
37 \( 1 + (0.200 + 0.979i)T \)
41 \( 1 + (-0.278 + 0.960i)T \)
43 \( 1 + (-0.200 + 0.979i)T \)
47 \( 1 + (-0.568 - 0.822i)T \)
53 \( 1 + (0.120 - 0.992i)T \)
59 \( 1 + (-0.987 + 0.160i)T \)
61 \( 1 + (0.799 - 0.600i)T \)
67 \( 1 + (0.996 - 0.0804i)T \)
71 \( 1 + (-0.692 - 0.721i)T \)
73 \( 1 + (-0.885 - 0.464i)T \)
79 \( 1 + (0.568 + 0.822i)T \)
83 \( 1 + (0.970 + 0.239i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.632 - 0.774i)T \)
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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

−27.69240502804788117266639000353, −26.77379105248243695694826806921, −26.27298669712892976450638110444, −25.14894505783242559543016509630, −24.51938825121086488509204466907, −23.01588165992819681980162223478, −22.208462815492792368535369390833, −21.66849078986810607256355834150, −20.39933798373657143719223078053, −19.06533401914182019413048790996, −17.78399048413321695077020457342, −17.35148987148997131983176181014, −15.874332277426099244951629079239, −15.14967475716338840014891798260, −14.368794613934130654578869446491, −13.71102733647538176342471577816, −11.81264619045806850696921369159, −10.629339611191582749933066648695, −9.48619270008529035176518934771, −8.67003266396184754688804220714, −7.37242890900109063544070933208, −6.21136004389416604080981821301, −4.97455412003842709963655216867, −4.03888951523353666024391841046, −2.45023201218521507426412249126, 1.17245989906944329568999434231, 1.86021598929081086315845949663, 3.57951627522919612105968575105, 4.871454997051219701848261955041, 6.15920215638905184463402768643, 7.98058550483053338403802681406, 8.643508021396738507472168799707, 9.76254680833346911023241645518, 11.34565558996412639578499564096, 11.959108145666779282269786424431, 13.14079447564181067310665015445, 13.780335533914852174723964470949, 14.679329012665885654406895992387, 16.76567080514561491516070017697, 17.56971278756949145689115724170, 18.35027271152870615192976379118, 19.64249205338111965968830641261, 20.15629487019559554702030774295, 21.07524753198393490165209737688, 22.098856702571833533988104124079, 23.440175997612481480559163388881, 24.19627873394354838798628119149, 24.93775175410333192884788107270, 26.328611818429190404705016182827, 27.410055335163969464537337822541

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