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Properties

Label 1-896-896.387-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: $\chi_{896} (387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ -0.969 + 0.244i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06091522745 + 0.4897058082i\)
\(L(\frac12)\) \(\approx\) \(0.06091522745 + 0.4897058082i\)
\(L(1)\) \(\approx\) \(1.099612324 + 0.1148043372i\)
\(L(1)\) \(\approx\) \(1.099612324 + 0.1148043372i\)

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

−21.52100388821009360730998637390, −20.24380236780006765595804512964, −19.88471560560426325672765602775, −19.023204405295649107621305009813, −18.24203034420978035669548292162, −17.708554078670875937303465417506, −16.79092408305350820275485847228, −15.38408660546719920919979870287, −14.828466779794861515390051564929, −14.36858329883666093169095217083, −13.37704227238148517964745585737, −12.52603518147169804026587646970, −11.87903933145730314450201608845, −10.70091879327329581973897568846, −9.85465111593249698406279730233, −9.170071506549773899162772097545, −7.870829049721053101141872746775, −7.40247867932611346823773631434, −6.628803787128591872477486435356, −5.657112964460569932727725475232, −4.26744104427069526913025654529, −3.28699683356097210312445663356, −2.44581580161653300476386624144, −1.645912649737628967305688055459, −0.08604588728058754096029207802, 1.3831205773257981239332622182, 2.38368875915472561956540923644, 3.64445358860756995828971175298, 4.2450904643614921887115422066, 5.29697530493167102465072808030, 6.05001570429935516269042189477, 7.69990636591965740360325062958, 8.076680602428270895892932634809, 9.30770459396175919464406183617, 9.46910755517017871165861020633, 10.60181857427371380995363160652, 11.64585402875294347095726592117, 12.48066337404302420949499353151, 13.40034179955576115983124299034, 14.22660464954487004317007926199, 14.75269205402734281143638384887, 15.902956487641112687189644759357, 16.69486014202498393431065776161, 16.78757139557210402348840187434, 18.31501753836921733316709664921, 19.24521394732632326686686668009, 19.747906216126590352535604439987, 20.67619215984843958148475080720, 21.206476198096230764266107532602, 21.87117633981195333095686442735

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