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Properties

Label 1-2669-2669.724-r0-0-0
Degree $1$
Conductor $2669$
Sign $-0.739 - 0.673i$
Analytic cond. $12.3947$
Root an. cond. $12.3947$
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  + (0.954 − 0.297i)2-s + (0.670 − 0.741i)3-s + (0.822 − 0.568i)4-s + (0.993 − 0.110i)5-s + (0.419 − 0.907i)6-s + (−0.542 − 0.839i)7-s + (0.616 − 0.787i)8-s + (−0.100 − 0.994i)9-s + (0.915 − 0.401i)10-s + (−0.907 − 0.419i)11-s + (0.130 − 0.991i)12-s + (0.866 − 0.5i)13-s + (−0.768 − 0.640i)14-s + (0.584 − 0.811i)15-s + (0.354 − 0.935i)16-s + ⋯
L(s)  = 1  + (0.954 − 0.297i)2-s + (0.670 − 0.741i)3-s + (0.822 − 0.568i)4-s + (0.993 − 0.110i)5-s + (0.419 − 0.907i)6-s + (−0.542 − 0.839i)7-s + (0.616 − 0.787i)8-s + (−0.100 − 0.994i)9-s + (0.915 − 0.401i)10-s + (−0.907 − 0.419i)11-s + (0.130 − 0.991i)12-s + (0.866 − 0.5i)13-s + (−0.768 − 0.640i)14-s + (0.584 − 0.811i)15-s + (0.354 − 0.935i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2669 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2669 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2669\)    =    \(17 \cdot 157\)
Sign: $-0.739 - 0.673i$
Analytic conductor: \(12.3947\)
Root analytic conductor: \(12.3947\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2669} (724, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2669,\ (0:\ ),\ -0.739 - 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.707155001 - 4.410016412i\)
\(L(\frac12)\) \(\approx\) \(1.707155001 - 4.410016412i\)
\(L(1)\) \(\approx\) \(2.002449914 - 1.675164060i\)
\(L(1)\) \(\approx\) \(2.002449914 - 1.675164060i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
157 \( 1 \)
good2 \( 1 + (0.954 - 0.297i)T \)
3 \( 1 + (0.670 - 0.741i)T \)
5 \( 1 + (0.993 - 0.110i)T \)
7 \( 1 + (-0.542 - 0.839i)T \)
11 \( 1 + (-0.907 - 0.419i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.373 + 0.927i)T \)
23 \( 1 + (0.685 - 0.728i)T \)
29 \( 1 + (0.150 + 0.988i)T \)
31 \( 1 + (-0.981 + 0.190i)T \)
37 \( 1 + (-0.780 + 0.624i)T \)
41 \( 1 + (0.999 + 0.0302i)T \)
43 \( 1 + (-0.941 + 0.335i)T \)
47 \( 1 + (0.721 - 0.692i)T \)
53 \( 1 + (-0.140 + 0.990i)T \)
59 \( 1 + (0.787 - 0.616i)T \)
61 \( 1 + (-0.714 + 0.699i)T \)
67 \( 1 + (-0.120 - 0.992i)T \)
71 \( 1 + (0.860 + 0.508i)T \)
73 \( 1 + (-0.670 + 0.741i)T \)
79 \( 1 + (0.995 - 0.0904i)T \)
83 \( 1 + (0.551 + 0.834i)T \)
89 \( 1 + (0.316 + 0.948i)T \)
97 \( 1 + (-0.401 + 0.915i)T \)
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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

−19.75394294794742659121812106405, −19.00476271448378139090159723119, −18.10904532063135744291883370479, −17.32959056210469619148028735193, −16.412601941201280975884640313571, −15.79552292182539597556001464860, −15.37044995261420911337288268705, −14.63777893491278444747561422859, −13.87829312573679434688111588570, −13.204363423251763799686677953568, −12.95267980025755774227801797573, −11.737493984783316557476307207009, −10.95275841501457592094641112349, −10.26786221217936204765183563188, −9.28403717542382868670282035473, −8.93412314384418729233990628114, −7.85720263131815058703775336530, −7.02563999351747523848384233611, −6.13321843941938400656537469932, −5.396470907306234832709947469308, −4.95018948841965252751823012569, −3.86387782606308611885469203361, −3.07569409254142588361706946798, −2.43062930148382086433419646136, −1.81343402984920934146140417670, 0.91320469754682705341891454558, 1.52427138906135329219555844235, 2.56428850843199420912933769937, 3.203545688314516784925783871163, 3.812470225502117790048265806806, 5.07436403954532914842979371286, 5.77355026034724972145877328765, 6.48368823276886469655407166307, 7.11040246930115527946288741755, 7.98843000623309601936776973838, 8.89019153872805844033845424286, 9.82669009866318308134164053376, 10.53198337929256052840049807976, 11.0218902144275730410498183983, 12.46464373800077549075058181491, 12.68090416333694309240976599735, 13.56514285773398950283420013436, 13.700222313818355650179077400309, 14.48072901209332834335729998692, 15.25346573593816856917259257192, 16.241564524789221654526174983752, 16.697924309670556380476824683106, 17.86874683072183613487558383131, 18.50297074835736596902905392368, 19.050430829870966477573719839804

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