Properties

Label 1-1045-1045.217-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.993 + 0.116i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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  + (−0.207 − 0.978i)2-s + (−0.406 + 0.913i)3-s + (−0.913 + 0.406i)4-s + (0.978 + 0.207i)6-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s i·12-s + (0.743 − 0.669i)13-s + (0.913 + 0.406i)14-s + (0.669 − 0.743i)16-s + (−0.743 − 0.669i)17-s + (−0.587 + 0.809i)18-s + (−0.5 − 0.866i)21-s + (0.866 + 0.5i)23-s + (−0.978 + 0.207i)24-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (−0.406 + 0.913i)3-s + (−0.913 + 0.406i)4-s + (0.978 + 0.207i)6-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s i·12-s + (0.743 − 0.669i)13-s + (0.913 + 0.406i)14-s + (0.669 − 0.743i)16-s + (−0.743 − 0.669i)17-s + (−0.587 + 0.809i)18-s + (−0.5 − 0.866i)21-s + (0.866 + 0.5i)23-s + (−0.978 + 0.207i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.993 + 0.116i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.993 + 0.116i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01137036170 + 0.1945469845i\)
\(L(\frac12)\) \(\approx\) \(0.01137036170 + 0.1945469845i\)
\(L(1)\) \(\approx\) \(0.6420708924 + 0.01585429887i\)
\(L(1)\) \(\approx\) \(0.6420708924 + 0.01585429887i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.207 - 0.978i)T \)
3 \( 1 + (-0.406 + 0.913i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
13 \( 1 + (0.743 - 0.669i)T \)
17 \( 1 + (-0.743 - 0.669i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.913 + 0.406i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (0.994 - 0.104i)T \)
53 \( 1 + (-0.743 + 0.669i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (0.978 + 0.207i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (-0.994 - 0.104i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (-0.951 + 0.309i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.207 + 0.978i)T \)
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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

−20.94995865749387712399522286044, −19.8818352424409767918968601257, −19.04564413206656139980360946683, −18.71286351856274095064707066877, −17.57937163477713515484812474454, −17.165472877042489592538957388652, −16.396102759832572893421659727586, −15.72280254003409335042026416847, −14.55936915170188651847024519171, −13.875767312174188290338753959626, −13.06967525601588528661455364114, −12.71323134590134389899667272645, −11.20350240613975116816040217198, −10.689682945365772894004274963895, −9.454505073783418897548293165547, −8.70939932550539724820277659257, −7.723130258661130928838467439948, −7.05891844617093385800542496134, −6.35543220505180801040790791205, −5.74628039083353230816298897627, −4.51185897816046415339378124959, −3.684307140638907370653692419378, −2.05968808377089151668886070023, −0.9175988851241992621404300675, −0.06312849245190422618629520105, 1.12471666875595628166851451969, 2.61496028182618556524172710555, 3.25502956805343191980914502085, 4.147174203461774644632154281613, 5.20658255931803171234718163800, 5.78266080576474949943702951887, 7.089887731560934775045400776561, 8.53491698733248300857909815759, 9.05493970223664548661141944874, 9.7123216242913402544111288283, 10.69228940727065347413875929117, 11.18201693178808931746962489661, 12.057878554028242356059972897978, 12.81075464143029821034367656768, 13.62647885612795014633574860582, 14.7218041778915166103967517440, 15.60015625828959215610528132377, 16.189895100983688832405240361502, 17.21711344922002786378404494753, 17.90221847450269199782309472436, 18.627923719191057645639864471814, 19.482588129528805675567083401201, 20.39719257712356579755138260342, 20.846961848727135163031184236235, 21.78724944327482356505376510561

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