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Properties

Label 1-448-448.387-r1-0-0
Degree $1$
Conductor $448$
Sign $-0.974 - 0.225i$
Analytic cond. $48.1442$
Root an. cond. $48.1442$
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  + (−0.793 + 0.608i)3-s + (0.608 − 0.793i)5-s + (0.258 − 0.965i)9-s + (0.130 − 0.991i)11-s + (0.382 − 0.923i)13-s + i·15-s + (−0.866 − 0.5i)17-s + (−0.991 + 0.130i)19-s + (0.258 − 0.965i)23-s + (−0.258 − 0.965i)25-s + (0.382 + 0.923i)27-s + (0.923 + 0.382i)29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + (−0.608 + 0.793i)37-s + ⋯
L(s)  = 1  + (−0.793 + 0.608i)3-s + (0.608 − 0.793i)5-s + (0.258 − 0.965i)9-s + (0.130 − 0.991i)11-s + (0.382 − 0.923i)13-s + i·15-s + (−0.866 − 0.5i)17-s + (−0.991 + 0.130i)19-s + (0.258 − 0.965i)23-s + (−0.258 − 0.965i)25-s + (0.382 + 0.923i)27-s + (0.923 + 0.382i)29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + (−0.608 + 0.793i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.974 - 0.225i$
Analytic conductor: \(48.1442\)
Root analytic conductor: \(48.1442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 448,\ (1:\ ),\ -0.974 - 0.225i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06968892009 - 0.6113609578i\)
\(L(\frac12)\) \(\approx\) \(0.06968892009 - 0.6113609578i\)
\(L(1)\) \(\approx\) \(0.7751965016 - 0.1567665218i\)
\(L(1)\) \(\approx\) \(0.7751965016 - 0.1567665218i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.793 + 0.608i)T \)
5 \( 1 + (0.608 - 0.793i)T \)
11 \( 1 + (0.130 - 0.991i)T \)
13 \( 1 + (0.382 - 0.923i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.991 + 0.130i)T \)
23 \( 1 + (0.258 - 0.965i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.608 + 0.793i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (0.923 - 0.382i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.130 + 0.991i)T \)
59 \( 1 + (0.991 + 0.130i)T \)
61 \( 1 + (0.130 + 0.991i)T \)
67 \( 1 + (-0.793 + 0.608i)T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (-0.965 + 0.258i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (0.382 - 0.923i)T \)
89 \( 1 + (0.965 + 0.258i)T \)
97 \( 1 - T \)
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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

−24.01772650690815178866129358633, −23.30334393152633859773126546488, −22.5650616154892525961586990275, −21.75208801795487907893625244766, −21.047900587540529269090901384350, −19.56629436214460414485709597924, −19.01467701013669576376818817371, −17.8881376780480644507108992051, −17.58859421152534877726687194781, −16.64662776369579201735050437865, −15.47211189632858416094213757375, −14.57089699174460851956154557053, −13.51411263941130453592700647766, −12.8895992166123214720755598095, −11.7308313585207196745746074975, −11.015956583874717684280728810793, −10.155957808668392963865903735, −9.12108244373222943306195984968, −7.74500493927617479461042444146, −6.68354338162050353099652303732, −6.3618797323238605672731799344, −5.09016453485126111551274316689, −3.98779306236732200941455002910, −2.26964590993270487591902394480, −1.6550079320399558315534557634, 0.18631467069613899596390620893, 1.211820815295319567508067881153, 2.88176078797620571113600700718, 4.20031464903645057926099409556, 5.09452697858996649076318269246, 5.922972465470135381650604534346, 6.73919303361026223310015731899, 8.534054248594614208105867445405, 8.937706251741660551653160026927, 10.30418693432865879939776648390, 10.75880453792823663660401061864, 11.9414775071118853160362038547, 12.78635592609374683522574693340, 13.6355387479729269980732087979, 14.783447872166688106573060671004, 15.92301526623266434828649486021, 16.36939784497122214696933614825, 17.380101702910746425876938955, 17.87925916332678649224573986007, 19.04310206031531213901217732959, 20.31116315089633931492614742426, 20.87324604452867310722478381398, 21.76683290369400322250578323557, 22.3710837937013427502913883381, 23.4087583656326892469840932940

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