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Properties

Label 1-896-896.565-r1-0-0
Degree $1$
Conductor $896$
Sign $-0.650 + 0.759i$
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  + (−0.321 + 0.946i)3-s + (−0.442 + 0.896i)5-s + (−0.793 − 0.608i)9-s + (−0.751 − 0.659i)11-s + (0.555 − 0.831i)13-s + (−0.707 − 0.707i)15-s + (0.258 + 0.965i)17-s + (0.0654 − 0.997i)19-s + (0.608 − 0.793i)23-s + (−0.608 − 0.793i)25-s + (0.831 − 0.555i)27-s + (0.195 + 0.980i)29-s + (0.866 + 0.5i)31-s + (0.866 − 0.5i)33-s + (−0.896 − 0.442i)37-s + ⋯
L(s)  = 1  + (−0.321 + 0.946i)3-s + (−0.442 + 0.896i)5-s + (−0.793 − 0.608i)9-s + (−0.751 − 0.659i)11-s + (0.555 − 0.831i)13-s + (−0.707 − 0.707i)15-s + (0.258 + 0.965i)17-s + (0.0654 − 0.997i)19-s + (0.608 − 0.793i)23-s + (−0.608 − 0.793i)25-s + (0.831 − 0.555i)27-s + (0.195 + 0.980i)29-s + (0.866 + 0.5i)31-s + (0.866 − 0.5i)33-s + (−0.896 − 0.442i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.650 + 0.759i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ -0.650 + 0.759i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4755702876 + 1.032782327i\)
\(L(\frac12)\) \(\approx\) \(0.4755702876 + 1.032782327i\)
\(L(1)\) \(\approx\) \(0.7608451850 + 0.3390811064i\)
\(L(1)\) \(\approx\) \(0.7608451850 + 0.3390811064i\)

Euler product

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

−21.14451044235355301625906361850, −20.74889673422520644660358163445, −19.827641113433297401559413968294, −18.943309031741415509419634156790, −18.47587895622850537594434335718, −17.43475765211249178753240500496, −16.80971740186594966542134433685, −15.98895917675201948737951145748, −15.22235024179187307391022115470, −13.828561908224166435780590977356, −13.48402271468626509375069591227, −12.42117177254559586728096235000, −11.946244022464146346578669293264, −11.20346930413982270450893123780, −9.987943821902055570754751158154, −9.019688211433877791941142872155, −8.07089404905343309245152327662, −7.509587300894215728368431253763, −6.55206918623206674955664495613, −5.47494037579868600551263491824, −4.81286431430742904490114083842, −3.63823494571859663782400264627, −2.29057189267591980754622966046, −1.372471937952800117157775778270, −0.358883858642913009905942413142, 0.80293323626417968551786542120, 2.77587727476420372997921075504, 3.24105348312736518492821398390, 4.2685308936496540847852027155, 5.296311481521956816145999704833, 6.106338162343586133458948528601, 7.03174072660312860955130404254, 8.24148137412333318994109633373, 8.80459046705199908255650564532, 10.21151936165740000562962694290, 10.62635621708178465800666646569, 11.17641827526574443836566744441, 12.203351952612493081030557213008, 13.22912162974516119092698278422, 14.24091208059772930281886234067, 15.05559440872829235822462861707, 15.61296696317249415066946942026, 16.26434182666789212355449810041, 17.300828043624700350398384390911, 18.03685563614721981645180043601, 18.84318732966121650852811704847, 19.74163678140741907571213935552, 20.528839689567314073364407156518, 21.53024621214213606208117946590, 21.85343453337330798939028996042

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