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Properties

Label 1-896-896.235-r1-0-0
Degree $1$
Conductor $896$
Sign $0.969 + 0.244i$
Analytic cond. $96.2885$
Root an. cond. $96.2885$
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  + (−0.751 + 0.659i)3-s + (−0.0654 − 0.997i)5-s + (0.130 − 0.991i)9-s + (−0.321 − 0.946i)11-s + (0.831 + 0.555i)13-s + (0.707 + 0.707i)15-s + (0.965 + 0.258i)17-s + (0.442 + 0.896i)19-s + (−0.991 − 0.130i)23-s + (−0.991 + 0.130i)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 + (−0.997 + 0.0654i)37-s + ⋯
L(s)  = 1  + (−0.751 + 0.659i)3-s + (−0.0654 − 0.997i)5-s + (0.130 − 0.991i)9-s + (−0.321 − 0.946i)11-s + (0.831 + 0.555i)13-s + (0.707 + 0.707i)15-s + (0.965 + 0.258i)17-s + (0.442 + 0.896i)19-s + (−0.991 − 0.130i)23-s + (−0.991 + 0.130i)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 + (−0.997 + 0.0654i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.969 + 0.244i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ 0.969 + 0.244i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.429547551 + 0.1778235274i\)
\(L(\frac12)\) \(\approx\) \(1.429547551 + 0.1778235274i\)
\(L(1)\) \(\approx\) \(0.8721210406 + 0.02393960276i\)
\(L(1)\) \(\approx\) \(0.8721210406 + 0.02393960276i\)

Euler product

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

−22.13883422602287289602318151289, −20.95452008897236059351059764848, −20.05635935909806355727070681085, −19.129440062966172681398849167617, −18.41510495374646089119439544251, −17.84954071783270835880233370412, −17.2897694133063235737945214449, −15.96709048209593257514335909958, −15.56765811628412056168820367111, −14.34135541720148930821214222814, −13.72007626109895444439949124880, −12.72605423137992672228948073984, −12.01912497944572829286544695973, −11.16546927552249331887168660529, −10.46496048263140236125438988784, −9.74070372043855181487184597175, −8.27212316967978896418348273564, −7.41974690867447759211833626170, −6.8900265858826564251176195998, −5.88192156206901571724461552059, −5.17974697023097535033173454581, −3.88659096259260565869966666377, −2.76780386133767945276719117292, −1.81264983193930585599366454233, −0.55495682471213203373760943163, 0.67728079951538192804372377431, 1.56225753523042772672281292730, 3.40057338997289830104605384539, 4.00677450417390757376006521155, 5.13222121581534246972130206522, 5.718592644081862682726878273074, 6.529450860276169634933762321490, 8.03565625880682901900765413784, 8.605121145682937392832074469419, 9.65114555471471028333178048140, 10.318081275786985384502615002154, 11.34688926011760011235110952760, 11.9746709636098164606701213474, 12.74503325917405517765420346150, 13.765205505100176521315888863976, 14.57929661357353783672051870341, 15.861324753610594439406117299635, 16.27552671656488378791103617808, 16.67520577372529447969190612397, 17.771873763721751736981521641501, 18.47454624644516173828562678303, 19.46964717487096709606472483514, 20.4523774449453179430483149634, 21.23448241471520493515179973729, 21.46119434678582582466828214398

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