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Properties

Label 1-13e2-169.25-r0-0-0
Degree $1$
Conductor $169$
Sign $-0.760 + 0.649i$
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  + (0.748 + 0.663i)2-s + (−0.354 + 0.935i)3-s + (0.120 + 0.992i)4-s + (0.970 + 0.239i)5-s + (−0.885 + 0.464i)6-s + (−0.568 + 0.822i)7-s + (−0.568 + 0.822i)8-s + (−0.748 − 0.663i)9-s + (0.568 + 0.822i)10-s + (0.748 − 0.663i)11-s + (−0.970 − 0.239i)12-s + (−0.970 + 0.239i)14-s + (−0.568 + 0.822i)15-s + (−0.970 + 0.239i)16-s + (0.568 − 0.822i)17-s + (−0.120 − 0.992i)18-s + ⋯
L(s)  = 1  + (0.748 + 0.663i)2-s + (−0.354 + 0.935i)3-s + (0.120 + 0.992i)4-s + (0.970 + 0.239i)5-s + (−0.885 + 0.464i)6-s + (−0.568 + 0.822i)7-s + (−0.568 + 0.822i)8-s + (−0.748 − 0.663i)9-s + (0.568 + 0.822i)10-s + (0.748 − 0.663i)11-s + (−0.970 − 0.239i)12-s + (−0.970 + 0.239i)14-s + (−0.568 + 0.822i)15-s + (−0.970 + 0.239i)16-s + (0.568 − 0.822i)17-s + (−0.120 − 0.992i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.760 + 0.649i$
Analytic conductor: \(0.784832\)
Root analytic conductor: \(0.784832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ -0.760 + 0.649i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5488669101 + 1.488834150i\)
\(L(\frac12)\) \(\approx\) \(0.5488669101 + 1.488834150i\)
\(L(1)\) \(\approx\) \(0.9995782133 + 1.056271103i\)
\(L(1)\) \(\approx\) \(0.9995782133 + 1.056271103i\)

Euler product

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

−27.7367811077945556724185575161, −25.871365821738932344873918964727, −25.128034036546048762789256334337, −24.1113537094448165691400578266, −23.23095541473116944232374722605, −22.50018749642745248225729786315, −21.48336198443425000482625523647, −20.34193612397239336841694236809, −19.53626519617566043620239283980, −18.607724082278583310097975879050, −17.33186683608120621847941534067, −16.69214496416962086314745825263, −14.78632458939455931083548481770, −13.97257650952746435011543330127, −12.834945189275112231638885559382, −12.66354857143613075130083864483, −11.10968408319318904283664435776, −10.21313875614632377591957016151, −9.04302932027460773683961119197, −7.08981560347012374421431060667, −6.303894894239993455342501689091, −5.25112651347787829977524506150, −3.797906838103397959140852238686, −2.19572999824867675070525805765, −1.1946439562728630218396939068, 2.64125294165239164643381932276, 3.69654526958600501097600644657, 5.19666560513188131514864935361, 5.89315880484074041959399190215, 6.79449806527534866350637946718, 8.77903939562383372871485190127, 9.420652024701885522453846777154, 10.88209791835113341140354286047, 11.97607291348991476015265561710, 13.10948307410345644591755911224, 14.30356977614205494320657808247, 14.98676762160294788267396026686, 16.12504675771875820470448282085, 16.84095801112074209169198281704, 17.73738256479061376080605690338, 19.09160908137793667575560333534, 20.88628022929692447101194849652, 21.32413972298356473275923087284, 22.40248241709906693956680833826, 22.63381349754312802161543971149, 24.141846557401412778096005066301, 25.29349916566218618212875971926, 25.70852195298982905549296572380, 26.85462866528447370018770089937, 27.80922241539787191747506677717

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