Properties

Label 1-896-896.229-r1-0-0
Degree $1$
Conductor $896$
Sign $0.453 + 0.891i$
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  + (−0.442 − 0.896i)3-s + (−0.946 + 0.321i)5-s + (−0.608 + 0.793i)9-s + (0.0654 + 0.997i)11-s + (−0.195 − 0.980i)13-s + (0.707 + 0.707i)15-s + (−0.258 − 0.965i)17-s + (−0.659 − 0.751i)19-s + (−0.793 − 0.608i)23-s + (0.793 − 0.608i)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 + (−0.321 − 0.946i)37-s + ⋯
L(s)  = 1  + (−0.442 − 0.896i)3-s + (−0.946 + 0.321i)5-s + (−0.608 + 0.793i)9-s + (0.0654 + 0.997i)11-s + (−0.195 − 0.980i)13-s + (0.707 + 0.707i)15-s + (−0.258 − 0.965i)17-s + (−0.659 − 0.751i)19-s + (−0.793 − 0.608i)23-s + (0.793 − 0.608i)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 + (−0.321 − 0.946i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.453 + 0.891i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ 0.453 + 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2203231858 + 0.1351471190i\)
\(L(\frac12)\) \(\approx\) \(0.2203231858 + 0.1351471190i\)
\(L(1)\) \(\approx\) \(0.5975485460 - 0.1849666353i\)
\(L(1)\) \(\approx\) \(0.5975485460 - 0.1849666353i\)

Euler product

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

−21.565992891055268381546663486026, −21.01038286152399810015402981685, −20.05241590047928813164166052076, −19.28953162757487092340855375486, −18.626407266522788859956885482957, −17.32334456504195555375516240950, −16.68606420717123285079624569557, −16.152602467936988933833034325966, −15.315638221112706166005351710733, −14.64228365148446720812263263980, −13.66914534923859133671259634612, −12.46406429350389920466215465122, −11.771316363992939952745236656464, −11.097473193975963451866396051279, −10.34981363432073494723952175853, −9.24288840587136407770012982750, −8.57513986841380861226636305919, −7.72072920450199178820487812359, −6.3958257615630157777635093971, −5.738294965441943831515900027629, −4.49226570962520486457560789001, −4.01052539138398922190849960503, −3.13464883979251155246980539841, −1.496967038470941465664896182022, −0.09622258367748272216968697284, 0.63814383948518159773855599696, 2.12235044129866398688271162210, 2.91524271463680258434330780081, 4.28550211149641046695511430945, 5.07249234091161426822957776044, 6.26417690040631398964253593449, 7.07522770430050637195938071255, 7.6864765395123267533592197606, 8.44736221380307609829030148975, 9.706942162301514352037310246392, 10.790796885355136325143980420607, 11.37945721966654204597227835235, 12.37287555949877097496548619935, 12.698081517487534027763230530726, 13.81877556017570189354055238382, 14.72213385019815741273525651408, 15.52638375196100156127969893356, 16.28382254467222937291101358545, 17.41118750178831659648987408959, 17.922894532282737811552968752572, 18.65096359001594540429540294893, 19.68399935444660908313472345320, 19.90822959736331551991884312186, 20.99739535339810536277140420321, 22.38746813063146994415294704547

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