↑

Properties

Label 2-756-36.11-c1-0-19
Degree $2$
Conductor $756$
Sign $-0.846 + 0.532i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + (−0.412 − 1.35i)2-s + (−1.65 + 1.11i)4-s + (−2.51 + 1.45i)5-s + (0.866 + 0.5i)7-s + (2.19 + 1.78i)8-s + (3.00 + 2.80i)10-s + (−1.18 + 2.05i)11-s + (−0.125 − 0.218i)13-s + (0.318 − 1.37i)14-s + (1.50 − 3.70i)16-s − 7.60i·17-s − 2.60i·19-s + (2.55 − 5.22i)20-s + (3.26 + 0.754i)22-s + (−4.51 − 7.82i)23-s + ⋯
L(s)  = 1  + (−0.292 − 0.956i)2-s + (−0.829 + 0.558i)4-s + (−1.12 + 0.650i)5-s + (0.327 + 0.188i)7-s + (0.776 + 0.630i)8-s + (0.950 + 0.887i)10-s + (−0.357 + 0.618i)11-s + (−0.0349 − 0.0604i)13-s + (0.0851 − 0.368i)14-s + (0.375 − 0.926i)16-s − 1.84i·17-s − 0.598i·19-s + (0.570 − 1.16i)20-s + (0.695 + 0.160i)22-s + (−0.941 − 1.63i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.846 + 0.532i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.846 + 0.532i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.149544 - 0.518515i\)
\(L(\frac12)\) \(\approx\) \(0.149544 - 0.518515i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.412 + 1.35i)T \)
3 \( 1 \)
7 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (2.51 - 1.45i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.18 - 2.05i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.125 + 0.218i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.60iT - 17T^{2} \)
19 \( 1 + 2.60iT - 19T^{2} \)
23 \( 1 + (4.51 + 7.82i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.53 - 3.19i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.0905 - 0.0522i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.15T + 37T^{2} \)
41 \( 1 + (-7.24 + 4.18i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.18 + 4.14i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.248 - 0.430i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.38iT - 53T^{2} \)
59 \( 1 + (3.99 + 6.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.69 + 6.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.444 + 0.256i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.45T + 71T^{2} \)
73 \( 1 - 4.06T + 73T^{2} \)
79 \( 1 + (10.0 + 5.79i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.439 + 0.761i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 14.6iT - 89T^{2} \)
97 \( 1 + (-2.88 + 4.98i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.22810794062788503800926302061, −9.218431484803010744617695610630, −8.343264890118589919017450120439, −7.55719766846412075902480800445, −6.82857436518013493537292940441, −5.03954739451183943282563153067, −4.37519304794667456684398601695, −3.16947037078240881295197879582, −2.33278196224050713520196614750, −0.33456855763167607757701844368, 1.33626524945250610698836951299, 3.71071579328078928282849431459, 4.34412233704806094057579070205, 5.49523683425411455708314059595, 6.27481063524477116147699288356, 7.54659354173383593037324203990, 8.172033319967080422635658937520, 8.486718439397288885439229737631, 9.725688382409246081653817856419, 10.53455793443720048221912187038

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