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Properties

Label 1-896-896.501-r0-0-0
Degree $1$
Conductor $896$
Sign $0.244 - 0.969i$
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  + (−0.659 − 0.751i)3-s + (0.997 − 0.0654i)5-s + (−0.130 + 0.991i)9-s + (−0.946 + 0.321i)11-s + (0.555 − 0.831i)13-s + (−0.707 − 0.707i)15-s + (0.965 + 0.258i)17-s + (−0.896 + 0.442i)19-s + (−0.991 − 0.130i)23-s + (0.991 − 0.130i)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 + (−0.0654 − 0.997i)37-s + ⋯
L(s)  = 1  + (−0.659 − 0.751i)3-s + (0.997 − 0.0654i)5-s + (−0.130 + 0.991i)9-s + (−0.946 + 0.321i)11-s + (0.555 − 0.831i)13-s + (−0.707 − 0.707i)15-s + (0.965 + 0.258i)17-s + (−0.896 + 0.442i)19-s + (−0.991 − 0.130i)23-s + (0.991 − 0.130i)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 + (−0.0654 − 0.997i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.244 - 0.969i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ 0.244 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.005142326 - 0.7827444540i\)
\(L(\frac12)\) \(\approx\) \(1.005142326 - 0.7827444540i\)
\(L(1)\) \(\approx\) \(0.9516159335 - 0.3045279794i\)
\(L(1)\) \(\approx\) \(0.9516159335 - 0.3045279794i\)

Euler product

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

−21.92705487647533760987616475180, −21.23590110859037644682050083347, −21.03611661145281912834647713709, −19.917638614987319025009199290803, −18.57230612923073302050489826326, −18.22633032543281147156478747356, −17.21567819431167376935802273209, −16.58227790828276825686702256570, −15.92169011766556454425992246641, −14.98451033170157679501125105406, −14.08394780606501387093487733517, −13.36500374453261474203427198581, −12.353344802449133009999443781471, −11.47987300157732226467807396360, −10.51158410995655401563127557841, −10.12088439906611628675714878400, −9.1512539101999581965050746689, −8.35964996378014363734420419453, −6.92804611233367440244391738725, −6.12615316715528845145000573691, −5.41343461709134024868317502371, −4.611168931676226568030084225583, −3.48331719913949331683151489985, −2.42624782462917673029123960912, −1.126361117160819634165018487275, 0.69892113541100969717969280239, 1.92039416012128072587935001050, 2.602494067144099889152455021976, 4.12762604035181630041674230464, 5.4960693396800496076940044572, 5.73143515416614123074500664861, 6.668900119308028969668384890249, 7.82707757277784651410223685598, 8.32902681474753797870535671390, 9.82242340408920916916609353518, 10.3708112847205614677002685818, 11.144724709325076838109219730370, 12.43953103123684460012739239928, 12.7560419514074070237177243954, 13.62961593318099558248054070608, 14.33573329221633857126255855529, 15.52693121562211196208642386299, 16.357492654356412368084377462674, 17.36000648048111613626134637308, 17.64550913705612783104617167173, 18.60546605459596368415353954527, 19.06568153594610189788494306658, 20.38482204177568194492066246996, 21.01201861485266365126463818128, 21.788074225828379301839440485846

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