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Properties

Label 1-896-896.531-r0-0-0
Degree $1$
Conductor $896$
Sign $-0.941 - 0.336i$
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  + (0.555 + 0.831i)3-s + (0.195 + 0.980i)5-s + (−0.382 + 0.923i)9-s + (−0.831 − 0.555i)11-s + (−0.195 + 0.980i)13-s + (−0.707 + 0.707i)15-s + (−0.707 − 0.707i)17-s + (−0.980 − 0.195i)19-s + (−0.923 − 0.382i)23-s + (−0.923 + 0.382i)25-s + (−0.980 + 0.195i)27-s + (−0.831 + 0.555i)29-s − i·31-s − i·33-s + (0.980 − 0.195i)37-s + ⋯
L(s)  = 1  + (0.555 + 0.831i)3-s + (0.195 + 0.980i)5-s + (−0.382 + 0.923i)9-s + (−0.831 − 0.555i)11-s + (−0.195 + 0.980i)13-s + (−0.707 + 0.707i)15-s + (−0.707 − 0.707i)17-s + (−0.980 − 0.195i)19-s + (−0.923 − 0.382i)23-s + (−0.923 + 0.382i)25-s + (−0.980 + 0.195i)27-s + (−0.831 + 0.555i)29-s − i·31-s − i·33-s + (0.980 − 0.195i)37-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.941 - 0.336i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ -0.941 - 0.336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1277384368 + 0.7361747511i\)
\(L(\frac12)\) \(\approx\) \(-0.1277384368 + 0.7361747511i\)
\(L(1)\) \(\approx\) \(0.8002080043 + 0.5118993207i\)
\(L(1)\) \(\approx\) \(0.8002080043 + 0.5118993207i\)

Euler product

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

−21.26613057075355110765788939053, −20.57560636412361053153041708568, −19.94777355082706356182280087726, −19.33160733174897391037728225050, −18.23162839106182315002333622756, −17.62540413407961306715876045297, −16.99678769677466584279351453314, −15.7426188683186607466034568066, −15.173619168768878712149750932455, −14.17423759099121110871693431857, −13.17569662598830849531049024332, −12.80167689883023926253071189347, −12.20483564791134460469111780552, −10.94439906764922190057364277447, −9.92014589391721865016004619175, −9.08915432503943287205131563382, −8.05494729563480725104610113367, −7.85388820670588368729344791218, −6.49544073345298418409805791294, −5.69665084158786727902852965491, −4.670355172091743783934865607066, −3.60713585219508805027149149177, −2.32398086338710674061806416897, −1.69886092156635508059863640018, −0.27056094231487583971884017573, 2.18791627846611808604476299618, 2.628934392665722452363434653165, 3.78713744239796069564659906380, 4.551525750837032570330007635235, 5.687307028619130179838937736632, 6.62270856068483412894386268344, 7.62900416169535569756668369171, 8.49226326859413528028250266328, 9.460355785661615308420259253350, 10.09298989231839225092367018535, 11.09139972965495718080686657691, 11.39992498304555447326816055989, 13.045119764073661192707702714253, 13.68705998612500883105837424811, 14.57479171520431541801689067226, 15.00305872353647256768348044366, 16.07591955434708577031101937624, 16.524702134657713040524479779012, 17.74017707937181613004389497796, 18.56292922990086962106285708976, 19.20667756567968001618881242253, 20.06371453516908980078817525718, 20.96527965705044478671552496691, 21.650339814334327426291742290822, 22.12795811494385249408900980917

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