↑

Properties

Label 2-672-56.19-c1-0-14
Degree $2$
Conductor $672$
Sign $-0.987 + 0.157i$
Analytic cond. $5.36594$
Root an. cond. $2.31645$
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  + (0.866 − 0.5i)3-s + (−1.44 + 2.49i)5-s + (−2.63 + 0.194i)7-s + (0.499 − 0.866i)9-s + (−2.91 − 5.04i)11-s − 1.04·13-s + 2.88i·15-s + (−5.91 + 3.41i)17-s + (0.589 + 0.340i)19-s + (−2.18 + 1.48i)21-s + (−1.85 − 1.07i)23-s + (−1.65 − 2.86i)25-s − 0.999i·27-s − 6.61i·29-s + (−1.91 − 3.31i)31-s + ⋯
L(s)  = 1  + (0.499 − 0.288i)3-s + (−0.644 + 1.11i)5-s + (−0.997 + 0.0733i)7-s + (0.166 − 0.288i)9-s + (−0.878 − 1.52i)11-s − 0.290·13-s + 0.744i·15-s + (−1.43 + 0.827i)17-s + (0.135 + 0.0781i)19-s + (−0.477 + 0.324i)21-s + (−0.387 − 0.223i)23-s + (−0.331 − 0.573i)25-s − 0.192i·27-s − 1.22i·29-s + (−0.344 − 0.595i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.987 + 0.157i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.987 + 0.157i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00327724 - 0.0412997i\)
\(L(\frac12)\) \(\approx\) \(0.00327724 - 0.0412997i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (2.63 - 0.194i)T \)
good5 \( 1 + (1.44 - 2.49i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.91 + 5.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.04T + 13T^{2} \)
17 \( 1 + (5.91 - 3.41i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.589 - 0.340i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.85 + 1.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.61iT - 29T^{2} \)
31 \( 1 + (1.91 + 3.31i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.06 - 1.19i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.19iT - 41T^{2} \)
43 \( 1 + 1.34T + 43T^{2} \)
47 \( 1 + (5.52 - 9.57i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.99 - 4.03i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.81 - 3.93i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.63 - 2.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.65 - 11.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.08iT - 71T^{2} \)
73 \( 1 + (-4.88 + 2.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.9 + 6.32i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.482iT - 83T^{2} \)
89 \( 1 + (-10.7 - 6.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.63iT - 97T^{2} \)
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

−10.20879340503218255638977116771, −9.167934137636032077499619471961, −8.223926207183467450084410224173, −7.54845284623733524504599771713, −6.48691376296776806417575092252, −5.96590324896615982232358357925, −4.16968259281275193298634551850, −3.20259266223640103457538208886, −2.51460553361489296170596360422, −0.01898382238640947946945865193, 2.12360031665981547214818261111, 3.41198767821173440843944369919, 4.62062697212283743913791558937, 5.04619962769058684500404744065, 6.73552166926741448726429176292, 7.45968581324439468958551144664, 8.421308857890916284827859972937, 9.275722279926479278724316350339, 9.792095420798931997621970651021, 10.77669121227314707619701814398

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