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Properties

Label 1-896-896.283-r0-0-0
Degree $1$
Conductor $896$
Sign $0.976 - 0.215i$
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  + (−0.997 − 0.0654i)3-s + (0.751 + 0.659i)5-s + (0.991 + 0.130i)9-s + (0.896 + 0.442i)11-s + (−0.195 − 0.980i)13-s + (−0.707 − 0.707i)15-s + (0.965 + 0.258i)17-s + (0.321 − 0.946i)19-s + (0.130 − 0.991i)23-s + (0.130 + 0.991i)25-s + (−0.980 − 0.195i)27-s + (−0.831 − 0.555i)29-s + (0.866 − 0.5i)31-s + (−0.866 − 0.5i)33-s + (−0.659 + 0.751i)37-s + ⋯
L(s)  = 1  + (−0.997 − 0.0654i)3-s + (0.751 + 0.659i)5-s + (0.991 + 0.130i)9-s + (0.896 + 0.442i)11-s + (−0.195 − 0.980i)13-s + (−0.707 − 0.707i)15-s + (0.965 + 0.258i)17-s + (0.321 − 0.946i)19-s + (0.130 − 0.991i)23-s + (0.130 + 0.991i)25-s + (−0.980 − 0.195i)27-s + (−0.831 − 0.555i)29-s + (0.866 − 0.5i)31-s + (−0.866 − 0.5i)33-s + (−0.659 + 0.751i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.976 - 0.215i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ 0.976 - 0.215i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.326951038 - 0.1444487084i\)
\(L(\frac12)\) \(\approx\) \(1.326951038 - 0.1444487084i\)
\(L(1)\) \(\approx\) \(1.003875419 + 0.002784617277i\)
\(L(1)\) \(\approx\) \(1.003875419 + 0.002784617277i\)

Euler product

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

−21.82010474173441901286092743475, −21.37049666866513420958541029524, −20.68069582136367687893998552982, −19.470196926820296311397418424935, −18.758696546380687609657415815, −17.80957489919506466118133366319, −17.15630360914043798847752854394, −16.43338661768413483851467081901, −16.09614109794637018743091306116, −14.6080419415941623283760683777, −13.96949532870987124037168616098, −13.00675049929129391110537261380, −12.111618753209608784034166862404, −11.65361000116601305449222003351, −10.59899528989673413846636057457, −9.6004143159882087871805784630, −9.229584149693890149230064174156, −7.8909588074952001835958213266, −6.82885636773533446858134029628, −5.99876673885010350070674509739, −5.35976051894477081389863544056, −4.441050532803887873379208193770, −3.44176355490138880794066246370, −1.72045523000544615472930763785, −1.125770913779720166118772671784, 0.83736121720122222078999870266, 1.98815393397702512245790269871, 3.13264607494713128859325285582, 4.337901282855974229305981878188, 5.36514763317117711636718602652, 6.042924291749510699870635575862, 6.874443191065466182156856754401, 7.57980623519534420391558441470, 8.97555156385231928127213029939, 10.05128215634744384922103057173, 10.35357840989632027847081758989, 11.42487661455685470743838664390, 12.12982463560644986861736135037, 13.01683312410955839011605698504, 13.80874930600598476326118097156, 14.891285884772072381112293542345, 15.3962939112650419201946609711, 16.73100737429250171989385658149, 17.15805607117707886404190547233, 17.87345323309357261459219050304, 18.57300410522154650027856532160, 19.37590118189910010013925217544, 20.50008871644247134324673043661, 21.29474106882274921553240872249, 22.227894216752168486217193294263

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