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Properties

Label 1-896-896.299-r0-0-0
Degree $1$
Conductor $896$
Sign $0.759 + 0.650i$
Analytic cond. $4.16100$
Root an. cond. $4.16100$
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  + (0.946 + 0.321i)3-s + (0.896 + 0.442i)5-s + (0.793 + 0.608i)9-s + (0.659 − 0.751i)11-s + (0.831 + 0.555i)13-s + (0.707 + 0.707i)15-s + (0.258 + 0.965i)17-s + (−0.997 − 0.0654i)19-s + (0.608 − 0.793i)23-s + (0.608 + 0.793i)25-s + (0.555 + 0.831i)27-s + (−0.980 + 0.195i)29-s + (−0.866 − 0.5i)31-s + (0.866 − 0.5i)33-s + (−0.442 + 0.896i)37-s + ⋯
L(s)  = 1  + (0.946 + 0.321i)3-s + (0.896 + 0.442i)5-s + (0.793 + 0.608i)9-s + (0.659 − 0.751i)11-s + (0.831 + 0.555i)13-s + (0.707 + 0.707i)15-s + (0.258 + 0.965i)17-s + (−0.997 − 0.0654i)19-s + (0.608 − 0.793i)23-s + (0.608 + 0.793i)25-s + (0.555 + 0.831i)27-s + (−0.980 + 0.195i)29-s + (−0.866 − 0.5i)31-s + (0.866 − 0.5i)33-s + (−0.442 + 0.896i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.759 + 0.650i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ 0.759 + 0.650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.502303042 + 0.9243948451i\)
\(L(\frac12)\) \(\approx\) \(2.502303042 + 0.9243948451i\)
\(L(1)\) \(\approx\) \(1.738082896 + 0.3633552460i\)
\(L(1)\) \(\approx\) \(1.738082896 + 0.3633552460i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.946 + 0.321i)T \)
5 \( 1 + (0.896 + 0.442i)T \)
11 \( 1 + (0.659 - 0.751i)T \)
13 \( 1 + (0.831 + 0.555i)T \)
17 \( 1 + (0.258 + 0.965i)T \)
19 \( 1 + (-0.997 - 0.0654i)T \)
23 \( 1 + (0.608 - 0.793i)T \)
29 \( 1 + (-0.980 + 0.195i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (-0.442 + 0.896i)T \)
41 \( 1 + (-0.382 - 0.923i)T \)
43 \( 1 + (0.195 - 0.980i)T \)
47 \( 1 + (-0.965 - 0.258i)T \)
53 \( 1 + (0.659 - 0.751i)T \)
59 \( 1 + (-0.0654 - 0.997i)T \)
61 \( 1 + (-0.751 + 0.659i)T \)
67 \( 1 + (0.946 + 0.321i)T \)
71 \( 1 + (0.923 + 0.382i)T \)
73 \( 1 + (0.130 - 0.991i)T \)
79 \( 1 + (-0.258 + 0.965i)T \)
83 \( 1 + (-0.555 + 0.831i)T \)
89 \( 1 + (-0.991 + 0.130i)T \)
97 \( 1 - iT \)
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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

−21.50079596165144707219210181339, −21.073868308712391761719646086023, −20.219807695159396938153286793739, −19.73889817292283165682701050483, −18.59998778890968177467244923257, −18.01630044682113263530170246499, −17.21825064673802183873424511898, −16.29593694058605563074480841303, −15.28764206273827329558757653323, −14.56486546173626465958204200826, −13.8184455520296575488611121934, −13.00163254618180810501217239879, −12.57710910387916174639181122293, −11.33474504515508034269861532774, −10.21033801022576162249831747492, −9.34222465280630543994339692216, −8.94841392440342892669116781109, −7.88013303146373026259397252608, −6.995056485373670425874312391347, −6.11193966973714317581230023709, −5.05319880265286813172453213197, −3.98619893378171631166749993998, −2.995824964783464600427883757089, −1.917432016349344225386306252625, −1.2171424894652723406429639129, 1.509328340122821286827385240453, 2.21382322508426483277427728221, 3.429517193834604514362249454818, 3.98473755869870868797510580503, 5.32915935174003441148241307581, 6.33529411500309014744345505064, 7.01009289756316349979785759421, 8.42293881147252435626543359478, 8.80946473243064030206756714853, 9.71299942926400927601067740806, 10.62128833421079614693430590797, 11.18262403624390704625577245969, 12.68146580476479833848823505015, 13.358838635094565626237046386568, 14.110661000868826715102241064057, 14.71286790060036029676797971900, 15.422393346469035105650490681404, 16.7104543687874228718760835127, 16.97377131610627099844117742651, 18.42586542321883373446435487769, 18.845563201491782936367682532915, 19.592995620332568481481845253809, 20.71175078424804821308799850239, 21.16376092584698294759394979951, 21.90064909870573529570658621082

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