↑

Properties

Label 1-896-896.445-r0-0-0
Degree $1$
Conductor $896$
Sign $0.988 + 0.148i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.988 + 0.148i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ 0.988 + 0.148i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.429806437 + 0.1069602148i\)
\(L(\frac12)\) \(\approx\) \(1.429806437 + 0.1069602148i\)
\(L(1)\) \(\approx\) \(1.054459585 + 0.04793747630i\)
\(L(1)\) \(\approx\) \(1.054459585 + 0.04793747630i\)

Euler product

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

−21.84121943262243293344219630520, −21.53984405676933525246418618449, −20.44126803344854638139123633811, −19.37026138614593260551115726089, −18.46772882855087498881195904977, −18.05056932013215695164945704294, −17.29202583203709810177441981113, −16.329002153092321160371448154918, −15.94443169351296929111045029728, −14.51766028085345581236871596428, −13.68665360511128363361853291244, −13.334812283963806475708210635278, −11.99237552047334087171275371592, −11.4220811271760599747944698551, −10.72072992200405695251014844077, −9.69651397130198682164966257964, −9.03425097033053851477379562256, −7.655875200893193416999353361045, −6.80610612411144034456852084176, −6.032630783627424747054248649863, −5.49906344082770696048992902470, −4.30566292685357289685117784615, −3.14023197942594499682255386405, −1.88002860578209337016013655889, −0.98080693927074519024369459244, 1.0226817701332556966319286102, 1.84759195412568266503461231029, 3.40791015583327546789412574697, 4.47920850647333958781136791795, 5.20519395992748346760624767904, 6.194449151324380640772301024972, 6.61162046398315252745144209220, 8.01217545715118758592988125483, 9.06329036662297422707454805547, 9.84502814056768877834156567833, 10.503176287081281830301407055746, 11.377311385288295673019377741342, 12.44569181304354705110274611481, 12.82372490776601055524826864434, 13.93837918022642110351308617920, 14.80506984628763464331433648032, 15.84348909396546643336443582037, 16.41788149803927947562976527487, 17.36204755605703468537235729246, 17.80294347776814345214383021464, 18.44036394203284066846271796606, 19.820688863479322840143093175669, 20.51574306506692593594375466236, 21.26007515942073356580216806036, 22.1105878524413902510090648600

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