↑

Properties

Label 1-896-896.627-r1-0-0
Degree $1$
Conductor $896$
Sign $-0.240 - 0.970i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.240 - 0.970i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (627, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ -0.240 - 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.112246098 - 1.422167549i\)
\(L(\frac12)\) \(\approx\) \(1.112246098 - 1.422167549i\)
\(L(1)\) \(\approx\) \(0.9217672912 - 0.3755485128i\)
\(L(1)\) \(\approx\) \(0.9217672912 - 0.3755485128i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.896 - 0.442i)T \)
5 \( 1 + (0.321 - 0.946i)T \)
11 \( 1 + (0.997 + 0.0654i)T \)
13 \( 1 + (0.980 + 0.195i)T \)
17 \( 1 + (0.258 - 0.965i)T \)
19 \( 1 + (-0.751 - 0.659i)T \)
23 \( 1 + (-0.793 + 0.608i)T \)
29 \( 1 + (-0.555 - 0.831i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (0.946 + 0.321i)T \)
41 \( 1 + (0.923 + 0.382i)T \)
43 \( 1 + (0.831 + 0.555i)T \)
47 \( 1 + (-0.965 + 0.258i)T \)
53 \( 1 + (0.997 + 0.0654i)T \)
59 \( 1 + (0.659 + 0.751i)T \)
61 \( 1 + (0.0654 + 0.997i)T \)
67 \( 1 + (0.896 + 0.442i)T \)
71 \( 1 + (-0.382 - 0.923i)T \)
73 \( 1 + (0.991 - 0.130i)T \)
79 \( 1 + (0.258 + 0.965i)T \)
83 \( 1 + (0.195 - 0.980i)T \)
89 \( 1 + (0.130 - 0.991i)T \)
97 \( 1 - iT \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.96823007427586516554769859552, −21.45477317724756315023216354499, −20.61834345705448898004506507988, −19.47490060428261535877358392556, −18.65978896304294726511770790161, −17.95979397815237604123982813944, −17.24149993833262209963807891429, −16.499800983146986743749041795402, −15.66028614769347944208302902233, −14.71866239681812825848378216042, −14.2402971744622394174697019362, −12.98078091647996097413560808504, −12.19314187359675146104382622076, −11.23654330327035072366399469801, −10.65725410266801213361958329207, −10.0045042217702948295768901058, −9.01056790552445610679767531552, −7.93843130768532373526721976218, −6.55856712435904667446728624524, −6.32501918892427237594128773547, −5.439215853141435201548828341088, −3.99067277019588104895581144798, −3.64703003082161875809451640100, −2.08076378441629456274982774767, −0.97041295322276809673694789639, 0.57204699022760993841965243141, 1.27379747492340509364707403041, 2.3316529826305453100305939888, 4.05857629720920415767564569026, 4.65574203720189900488577576004, 5.85406757776322999687830541784, 6.24608269225303118148418019578, 7.381185262510485881415035539883, 8.332211734490956985383728132764, 9.30348705693187227860219785302, 10.0063408015644411077795394067, 11.40761854088797929070120352478, 11.600517021006720183415114876126, 12.671496890194771996641746009114, 13.36361343138985756543916525885, 14.001931888779986156168361547009, 15.33980478183759964328131100838, 16.268393719998774952700083019020, 16.71714651891785031473825337075, 17.588737299874680126964505427875, 18.109221521147512133891790337782, 19.18940399642181859154017317058, 19.83975241973636584058851164233, 20.92040490039063811750057219880, 21.46224561633863556925186996878

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