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Properties

Label 1-896-896.517-r1-0-0
Degree $1$
Conductor $896$
Sign $-0.941 + 0.336i$
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  + (−0.831 − 0.555i)3-s + (−0.980 − 0.195i)5-s + (0.382 + 0.923i)9-s + (−0.555 − 0.831i)11-s + (0.980 − 0.195i)13-s + (0.707 + 0.707i)15-s + (−0.707 + 0.707i)17-s + (0.195 + 0.980i)19-s + (−0.923 + 0.382i)23-s + (0.923 + 0.382i)25-s + (0.195 − 0.980i)27-s + (0.555 − 0.831i)29-s − i·31-s − i·33-s + (0.195 − 0.980i)37-s + ⋯
L(s)  = 1  + (−0.831 − 0.555i)3-s + (−0.980 − 0.195i)5-s + (0.382 + 0.923i)9-s + (−0.555 − 0.831i)11-s + (0.980 − 0.195i)13-s + (0.707 + 0.707i)15-s + (−0.707 + 0.707i)17-s + (0.195 + 0.980i)19-s + (−0.923 + 0.382i)23-s + (0.923 + 0.382i)25-s + (0.195 − 0.980i)27-s + (0.555 − 0.831i)29-s − i·31-s − i·33-s + (0.195 − 0.980i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.941 + 0.336i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ -0.941 + 0.336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04360881152 - 0.2513237734i\)
\(L(\frac12)\) \(\approx\) \(-0.04360881152 - 0.2513237734i\)
\(L(1)\) \(\approx\) \(0.6014101359 - 0.1636827321i\)
\(L(1)\) \(\approx\) \(0.6014101359 - 0.1636827321i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.831 - 0.555i)T \)
5 \( 1 + (-0.980 - 0.195i)T \)
11 \( 1 + (-0.555 - 0.831i)T \)
13 \( 1 + (0.980 - 0.195i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
19 \( 1 + (0.195 + 0.980i)T \)
23 \( 1 + (-0.923 + 0.382i)T \)
29 \( 1 + (0.555 - 0.831i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.195 - 0.980i)T \)
41 \( 1 + (-0.923 + 0.382i)T \)
43 \( 1 + (0.831 - 0.555i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (0.555 + 0.831i)T \)
59 \( 1 + (0.980 + 0.195i)T \)
61 \( 1 + (0.831 + 0.555i)T \)
67 \( 1 + (-0.831 - 0.555i)T \)
71 \( 1 + (0.382 - 0.923i)T \)
73 \( 1 + (0.382 + 0.923i)T \)
79 \( 1 + (-0.707 - 0.707i)T \)
83 \( 1 + (-0.195 - 0.980i)T \)
89 \( 1 + (0.923 + 0.382i)T \)
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

−22.38230113492758662578584310004, −21.52497408672841324620733047691, −20.46740300904823960579475830768, −20.113409270403569597998218550671, −18.89262031568474706162404927746, −18.04477598685900164296298183485, −17.640740412901976884692630604, −16.2928754503769679804594422814, −15.87864290121947311645750384614, −15.34927229054665747137766720778, −14.34777927819182132684183081640, −13.19773993739159833804786055460, −12.28457773403133863731969378662, −11.59359749267825480700096216054, −10.89144897057260711233991811848, −10.196566961649085147871702974888, −9.12013603299374185913231756321, −8.25669481004758214995046895817, −7.03638639631581343692238994994, −6.58740689966634237105870779769, −5.22364307423897852604879112304, −4.56612570607159439209558260653, −3.75245117345483158639352794444, −2.647997586115588435708099489146, −1.02533450600124795689192585053, 0.08904673602858955869590777228, 0.95412726766720756894337163012, 2.19103916363963734170430189033, 3.63439926093257487143194437981, 4.329112376148887421005431700244, 5.667243296909204935029244836228, 6.07897152661140459714885705672, 7.29800923781981437957385785866, 8.06679714724288942451808901645, 8.63086917954027599784941232336, 10.194456239124448706229867657145, 10.91700823039038631910673290033, 11.60047339511122062808296139768, 12.30141510227218318692919641108, 13.20744524307202147475675649416, 13.797524911882271938325962737651, 15.163096629312035693765108942517, 15.93429997857402744770529239490, 16.41918613479076657056975433380, 17.356418395383548851435700593815, 18.29171137234865667080688170049, 18.834068992739015765918753282369, 19.56749010608343415760554367678, 20.480737205804868433890321290875, 21.39888802711221793823564551565

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