↑

Properties

Label 2-896-112.19-c1-0-18
Degree $2$
Conductor $896$
Sign $0.967 - 0.254i$
Analytic cond. $7.15459$
Root an. cond. $2.67480$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.665 + 2.48i)3-s + (3.12 − 0.837i)5-s + (−1.56 − 2.13i)7-s + (−3.13 − 1.80i)9-s + (0.376 − 1.40i)11-s + (3.11 − 3.11i)13-s + 8.31i·15-s + (2.02 − 1.16i)17-s + (4.40 − 1.18i)19-s + (6.34 − 2.45i)21-s + (1.15 − 1.99i)23-s + (4.72 − 2.73i)25-s + (1.12 − 1.12i)27-s + (1.55 + 1.55i)29-s + (−3.88 − 6.73i)31-s + ⋯
L(s)  = 1  + (−0.384 + 1.43i)3-s + (1.39 − 0.374i)5-s + (−0.590 − 0.807i)7-s + (−1.04 − 0.602i)9-s + (0.113 − 0.423i)11-s + (0.863 − 0.863i)13-s + 2.14i·15-s + (0.490 − 0.283i)17-s + (1.01 − 0.270i)19-s + (1.38 − 0.536i)21-s + (0.240 − 0.416i)23-s + (0.945 − 0.546i)25-s + (0.216 − 0.216i)27-s + (0.288 + 0.288i)29-s + (−0.698 − 1.20i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.967 - 0.254i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.967 - 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71952 + 0.222506i\)
\(L(\frac12)\) \(\approx\) \(1.71952 + 0.222506i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (1.56 + 2.13i)T \)
good3 \( 1 + (0.665 - 2.48i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-3.12 + 0.837i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.376 + 1.40i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.11 + 3.11i)T - 13iT^{2} \)
17 \( 1 + (-2.02 + 1.16i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.40 + 1.18i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.15 + 1.99i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.55 - 1.55i)T + 29iT^{2} \)
31 \( 1 + (3.88 + 6.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.272 + 1.01i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.77T + 41T^{2} \)
43 \( 1 + (-7.12 - 7.12i)T + 43iT^{2} \)
47 \( 1 + (1.42 - 2.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.0 + 2.97i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.77 - 1.01i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.72 - 13.9i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.59 - 0.695i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.48T + 71T^{2} \)
73 \( 1 + (5.65 + 9.78i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.706 - 0.408i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.65 - 2.65i)T + 83iT^{2} \)
89 \( 1 + (2.40 - 4.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.86iT - 97T^{2} \)
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

−10.03330843731184060426897345893, −9.567250790840043121942669891018, −8.919625824120882370797229383832, −7.65852220993143019510058537353, −6.31498566575432479161586994885, −5.67870763105679549861539443819, −4.96125947689129462998022393502, −3.84479460420843950086639813738, −2.96270740460563417437027178198, −0.999928704110505210183670877016, 1.43323844771511603792869007699, 2.13346051164733935016495281706, 3.35999382795026696540808151818, 5.30142308220739619162767784990, 5.94608533126547422939238953673, 6.58611441474086505006988260502, 7.19642612989372858724852884863, 8.392768986525922365031728476467, 9.331210727155916898398306588069, 9.921390295366796897461355666995

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