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Properties

Label 1-896-896.213-r1-0-0
Degree $1$
Conductor $896$
Sign $-0.819 + 0.572i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.819 + 0.572i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ -0.819 + 0.572i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4144030611 - 1.317314814i\)
\(L(\frac12)\) \(\approx\) \(-0.4144030611 - 1.317314814i\)
\(L(1)\) \(\approx\) \(0.9641479784 - 0.6569966966i\)
\(L(1)\) \(\approx\) \(0.9641479784 - 0.6569966966i\)

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

−22.25569021107422845196518396690, −21.39577156006724484569449252963, −20.559540638548309343737391575042, −20.062689590573296695915783078550, −19.002623713226144797511929199866, −18.447880654649149230571616170426, −17.549720253364360132004460643406, −16.52403159451267346942237106074, −15.555649206283392921597759888225, −14.98408030782742788832148299924, −14.5528560206769564964020575331, −13.353976221681909551746065794057, −12.88410957141209495676679802238, −11.36133936496789359660746548054, −10.75886509973054116967205302174, −10.0476804807452361216693332058, −9.21895680293029165973305716460, −8.20273507463128358454699687134, −7.49644405870546729376074482427, −6.52368591502321993107256795512, −5.44882645994777467835240214372, −4.277060510653822472967258507273, −3.582918914073788449454390753260, −2.6073250317375380609020909336, −1.78720864294237064994787866320, 0.25519023956308860867649082329, 1.22058983152594469641562756609, 2.209621944745278312884872857297, 3.33205516592863662553244255485, 4.24774565850935329315069562265, 5.34319931023839617422029902130, 6.37574677116037913513988172467, 7.240386214022930075379806549635, 8.27827430596751576383518584680, 8.9186507684890019271583119751, 9.31168909880131801139559183379, 10.93016599602806719446407510934, 11.53754374162326493181708991594, 12.71388949828544295203580473285, 13.28176331480626236274693931777, 13.72541446382992860318389712403, 14.87291724998862931939718543963, 15.69895844340632752148004034797, 16.464447957860284089274814598681, 17.342433423544697772343471634556, 18.297979593071097143933712871279, 18.93328753258816862267548002160, 19.8153763626378337971279395228, 20.334342542337188632192818318136, 21.2148318554868250193141400667

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