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Properties

Label 1-1073-1073.10-r1-0-0
Degree $1$
Conductor $1073$
Sign $-0.980 - 0.195i$
Analytic cond. $115.309$
Root an. cond. $115.309$
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  + (−0.997 + 0.0747i)2-s + (−0.930 + 0.365i)3-s + (0.988 − 0.149i)4-s + (−0.826 + 0.563i)5-s + (0.900 − 0.433i)6-s + (0.365 + 0.930i)7-s + (−0.974 + 0.222i)8-s + (0.733 − 0.680i)9-s + (0.781 − 0.623i)10-s + (−0.974 − 0.222i)11-s + (−0.866 + 0.5i)12-s + (0.733 + 0.680i)13-s + (−0.433 − 0.900i)14-s + (0.563 − 0.826i)15-s + (0.955 − 0.294i)16-s + (0.866 − 0.5i)17-s + ⋯
L(s)  = 1  + (−0.997 + 0.0747i)2-s + (−0.930 + 0.365i)3-s + (0.988 − 0.149i)4-s + (−0.826 + 0.563i)5-s + (0.900 − 0.433i)6-s + (0.365 + 0.930i)7-s + (−0.974 + 0.222i)8-s + (0.733 − 0.680i)9-s + (0.781 − 0.623i)10-s + (−0.974 − 0.222i)11-s + (−0.866 + 0.5i)12-s + (0.733 + 0.680i)13-s + (−0.433 − 0.900i)14-s + (0.563 − 0.826i)15-s + (0.955 − 0.294i)16-s + (0.866 − 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1073 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1073 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1073\)    =    \(29 \cdot 37\)
Sign: $-0.980 - 0.195i$
Analytic conductor: \(115.309\)
Root analytic conductor: \(115.309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1073} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1073,\ (1:\ ),\ -0.980 - 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02902004593 + 0.2939402089i\)
\(L(\frac12)\) \(\approx\) \(0.02902004593 + 0.2939402089i\)
\(L(1)\) \(\approx\) \(0.4176222990 + 0.1554582715i\)
\(L(1)\) \(\approx\) \(0.4176222990 + 0.1554582715i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.997 + 0.0747i)T \)
3 \( 1 + (-0.930 + 0.365i)T \)
5 \( 1 + (-0.826 + 0.563i)T \)
7 \( 1 + (0.365 + 0.930i)T \)
11 \( 1 + (-0.974 - 0.222i)T \)
13 \( 1 + (0.733 + 0.680i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.149 - 0.988i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
31 \( 1 + (0.433 - 0.900i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.433 - 0.900i)T \)
47 \( 1 + (0.974 + 0.222i)T \)
53 \( 1 + (0.0747 + 0.997i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.930 + 0.365i)T \)
67 \( 1 + (-0.955 - 0.294i)T \)
71 \( 1 + (0.733 + 0.680i)T \)
73 \( 1 + (0.433 + 0.900i)T \)
79 \( 1 + (-0.294 + 0.955i)T \)
83 \( 1 + (0.365 - 0.930i)T \)
89 \( 1 + (-0.997 + 0.0747i)T \)
97 \( 1 + (0.781 - 0.623i)T \)
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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

−20.8316131519357793427412009666, −19.95993059152779196517568092447, −19.23070779917554023283829477129, −18.35145810083323659051941553565, −17.85955377819339085560036271077, −16.9584506435285346188814636087, −16.3557628863925180553188225252, −15.865619207398505685875280940665, −14.88857267173839264131001999206, −13.557374585861697986887145603999, −12.49373502633428946678392737402, −12.19944083846591365728189693502, −11.07359521135967938201300449411, −10.550437135811762085278555906426, −9.988488847713947777907855202844, −8.43826899401197098195737155790, −7.86381893824046028221253405519, −7.44076599254413823278872357354, −6.26346328209319527341470138061, −5.43691443868573710338240631150, −4.32447978306949899614597055484, −3.32399735165094162437366502261, −1.73795544587464922173247243489, −0.95586063825587590367556105238, −0.15047449823169279294663261964, 0.88217978608196068659125335895, 2.264015888467742521620678109846, 3.2317644848653420653878139335, 4.45949203240842511589486122341, 5.60033390908435014399452177305, 6.21017110126458212082280930931, 7.24809103522770937953788974476, 7.94789013476579993668656423532, 8.884774053740526955886801040502, 9.7581949426341762357481899151, 10.66708187769392689717318660743, 11.31649929101312549872895252950, 11.7668635428735095285702093556, 12.553099257460812869141177615252, 14.03925005424383841464119717868, 15.24646942741443786326586747385, 15.59251620409680801512535037010, 16.15351938126115148672416069682, 17.04503954388687069389268485288, 18.033890239256741313173635724484, 18.53735435473862587936694535202, 18.89815165855322845390965503032, 20.05822737961538911240498986554, 21.0404689524603603192090609979, 21.506698387225179699200227891389

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