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Properties

Label 1-1012-1012.159-r0-0-0
Degree $1$
Conductor $1012$
Sign $-0.753 - 0.657i$
Analytic cond. $4.69970$
Root an. cond. $4.69970$
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  + (0.564 + 0.825i)3-s + (0.362 + 0.931i)5-s + (−0.985 − 0.170i)7-s + (−0.362 + 0.931i)9-s + (−0.466 − 0.884i)13-s + (−0.564 + 0.825i)15-s + (−0.0855 − 0.996i)17-s + (−0.921 − 0.389i)19-s + (−0.415 − 0.909i)21-s + (−0.736 + 0.676i)25-s + (−0.974 + 0.226i)27-s + (−0.921 + 0.389i)29-s + (−0.941 + 0.336i)31-s + (−0.198 − 0.980i)35-s + (0.254 + 0.967i)37-s + ⋯
L(s)  = 1  + (0.564 + 0.825i)3-s + (0.362 + 0.931i)5-s + (−0.985 − 0.170i)7-s + (−0.362 + 0.931i)9-s + (−0.466 − 0.884i)13-s + (−0.564 + 0.825i)15-s + (−0.0855 − 0.996i)17-s + (−0.921 − 0.389i)19-s + (−0.415 − 0.909i)21-s + (−0.736 + 0.676i)25-s + (−0.974 + 0.226i)27-s + (−0.921 + 0.389i)29-s + (−0.941 + 0.336i)31-s + (−0.198 − 0.980i)35-s + (0.254 + 0.967i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $-0.753 - 0.657i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ -0.753 - 0.657i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1097331261 + 0.2928917970i\)
\(L(\frac12)\) \(\approx\) \(-0.1097331261 + 0.2928917970i\)
\(L(1)\) \(\approx\) \(0.7920471999 + 0.3695665079i\)
\(L(1)\) \(\approx\) \(0.7920471999 + 0.3695665079i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.564 + 0.825i)T \)
5 \( 1 + (0.362 + 0.931i)T \)
7 \( 1 + (-0.985 - 0.170i)T \)
13 \( 1 + (-0.466 - 0.884i)T \)
17 \( 1 + (-0.0855 - 0.996i)T \)
19 \( 1 + (-0.921 - 0.389i)T \)
29 \( 1 + (-0.921 + 0.389i)T \)
31 \( 1 + (-0.941 + 0.336i)T \)
37 \( 1 + (0.254 + 0.967i)T \)
41 \( 1 + (-0.254 + 0.967i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.696 - 0.717i)T \)
59 \( 1 + (-0.897 - 0.441i)T \)
61 \( 1 + (-0.610 - 0.791i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (-0.198 + 0.980i)T \)
73 \( 1 + (0.516 + 0.856i)T \)
79 \( 1 + (-0.466 - 0.884i)T \)
83 \( 1 + (-0.998 - 0.0570i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (0.998 - 0.0570i)T \)
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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

−21.11265894126369655447448144979, −20.20163095063813583878733700137, −19.52065899707920794291661156862, −19.00166392162848289548520204487, −18.15745595058681291587564592230, −16.950816721926138658621880192328, −16.81150183567167430598989466485, −15.576820207887265174811231246208, −14.738687507090944744067325361238, −13.85935234432164250819306420928, −13.050219155538592108922393982821, −12.59675599860879547900698392396, −11.95393915978059430122607920744, −10.6461281069105830767754085102, −9.426879023651865365840088778469, −9.13553800232512661350006306393, −8.20298290408882541995723182076, −7.28406453848776625641360636600, −6.28955087610093240687649669392, −5.76761529291036909758302571235, −4.34058770390855381773509748155, −3.52239208907168854800428066750, −2.20729407063329417753619832442, −1.659842921261762311989288403416, −0.10831322707551451518974287709, 2.055609255687157168576423575790, 3.038559076348365381922226094388, 3.39855998271005850837497892817, 4.67582327030176532212315376901, 5.598082267349938242922427820574, 6.64635245167062933557895586543, 7.390582590315175491172075277011, 8.448942362362163749210086240283, 9.52456903065340774697701114349, 9.917718278340171697460950497608, 10.71699843539796174235338960239, 11.46239935514027072682046489602, 12.893045333587911044234599416440, 13.40564512307727685861639304543, 14.398399351293125671851999483643, 14.997248090684541600679432836570, 15.67611818624562106946925626513, 16.54768089141007603942800841829, 17.279783009512638042957823272397, 18.3809441918914225314452546425, 19.00040184171850135310264534305, 20.016235207437743165400600074246, 20.29066206468753842220784822630, 21.65373747598450115847844719407, 21.90739129167703171643169440033

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