↑

Properties

Label 2-896-896.181-c0-0-0
Degree $2$
Conductor $896$
Sign $0.0490 + 0.998i$
Analytic cond. $0.447162$
Root an. cond. $0.668701$
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  + (0.831 − 0.555i)2-s + (0.382 − 0.923i)4-s + (0.195 − 0.980i)7-s + (−0.195 − 0.980i)8-s + (−0.980 + 0.195i)9-s + (−0.368 − 0.448i)11-s + (−0.382 − 0.923i)14-s + (−0.707 − 0.707i)16-s + (−0.707 + 0.707i)18-s + (−0.555 − 0.168i)22-s + (1.63 + 1.08i)23-s + (0.555 + 0.831i)25-s + (−0.831 − 0.555i)28-s + (1.36 + 1.11i)29-s + (−0.980 − 0.195i)32-s + ⋯
L(s)  = 1  + (0.831 − 0.555i)2-s + (0.382 − 0.923i)4-s + (0.195 − 0.980i)7-s + (−0.195 − 0.980i)8-s + (−0.980 + 0.195i)9-s + (−0.368 − 0.448i)11-s + (−0.382 − 0.923i)14-s + (−0.707 − 0.707i)16-s + (−0.707 + 0.707i)18-s + (−0.555 − 0.168i)22-s + (1.63 + 1.08i)23-s + (0.555 + 0.831i)25-s + (−0.831 − 0.555i)28-s + (1.36 + 1.11i)29-s + (−0.980 − 0.195i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0490 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0490 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.0490 + 0.998i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :0),\ 0.0490 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.478321381\)
\(L(\frac12)\) \(\approx\) \(1.478321381\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.831 + 0.555i)T \)
7 \( 1 + (-0.195 + 0.980i)T \)
good3 \( 1 + (0.980 - 0.195i)T^{2} \)
5 \( 1 + (-0.555 - 0.831i)T^{2} \)
11 \( 1 + (0.368 + 0.448i)T + (-0.195 + 0.980i)T^{2} \)
13 \( 1 + (0.555 - 0.831i)T^{2} \)
17 \( 1 + (0.707 - 0.707i)T^{2} \)
19 \( 1 + (0.831 + 0.555i)T^{2} \)
23 \( 1 + (-1.63 - 1.08i)T + (0.382 + 0.923i)T^{2} \)
29 \( 1 + (-1.36 - 1.11i)T + (0.195 + 0.980i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (1.21 - 0.368i)T + (0.831 - 0.555i)T^{2} \)
41 \( 1 + (0.382 + 0.923i)T^{2} \)
43 \( 1 + (-0.151 + 1.53i)T + (-0.980 - 0.195i)T^{2} \)
47 \( 1 + (-0.707 + 0.707i)T^{2} \)
53 \( 1 + (1.53 - 1.26i)T + (0.195 - 0.980i)T^{2} \)
59 \( 1 + (0.555 + 0.831i)T^{2} \)
61 \( 1 + (-0.980 + 0.195i)T^{2} \)
67 \( 1 + (-0.195 + 0.0192i)T + (0.980 - 0.195i)T^{2} \)
71 \( 1 + (-1.81 - 0.360i)T + (0.923 + 0.382i)T^{2} \)
73 \( 1 + (0.923 - 0.382i)T^{2} \)
79 \( 1 + (1.02 + 0.425i)T + (0.707 + 0.707i)T^{2} \)
83 \( 1 + (-0.831 - 0.555i)T^{2} \)
89 \( 1 + (-0.382 + 0.923i)T^{2} \)
97 \( 1 - iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.65823442587707948762851382159, −9.399254754729228032778185781286, −8.543570076065021221476439144057, −7.34985323226751008626163989104, −6.62627349239791526737330856441, −5.38602683808429497646378304161, −4.92324565305152166782143840204, −3.55379079451134514687669992723, −2.91001894200419053481511645409, −1.27951006728395171995675577235, 2.43506151330254005419652350088, 3.07903344317896446218396414558, 4.60966907019686886690505464596, 5.19715557782216498518284759109, 6.20643896130596978825909642262, 6.81342358654258500317096803153, 8.169609713746782393011148485695, 8.483911084941405481700434804750, 9.492389810774458915861275379378, 10.80621164507848130976667683789

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