Properties

Label 2-722-19.7-c1-0-19
Degree $2$
Conductor $722$
Sign $-0.174 + 0.984i$
Analytic cond. $5.76519$
Root an. cond. $2.40108$
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  + (0.5 + 0.866i)2-s + (−1.26 − 2.18i)3-s + (−0.499 + 0.866i)4-s + (1.22 + 2.13i)5-s + (1.26 − 2.18i)6-s − 2.79·7-s − 0.999·8-s + (−1.67 + 2.90i)9-s + (−1.22 + 2.13i)10-s − 1.67·11-s + 2.52·12-s + (3.17 − 5.49i)13-s + (−1.39 − 2.41i)14-s + (3.09 − 5.36i)15-s + (−0.5 − 0.866i)16-s + (−2.48 − 4.30i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.727 − 1.26i)3-s + (−0.249 + 0.433i)4-s + (0.549 + 0.952i)5-s + (0.514 − 0.891i)6-s − 1.05·7-s − 0.353·8-s + (−0.558 + 0.967i)9-s + (−0.388 + 0.673i)10-s − 0.506·11-s + 0.727·12-s + (0.879 − 1.52i)13-s + (−0.373 − 0.646i)14-s + (0.800 − 1.38i)15-s + (−0.125 − 0.216i)16-s + (−0.602 − 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.174 + 0.984i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ -0.174 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.520422 - 0.620691i\)
\(L(\frac12)\) \(\approx\) \(0.520422 - 0.620691i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 \)
good3 \( 1 + (1.26 + 2.18i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.22 - 2.13i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 2.79T + 7T^{2} \)
11 \( 1 + 1.67T + 11T^{2} \)
13 \( 1 + (-3.17 + 5.49i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.48 + 4.30i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.24 + 2.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.96 + 5.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.28T + 31T^{2} \)
37 \( 1 + 0.550T + 37T^{2} \)
41 \( 1 + (1.30 + 2.25i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.43 + 2.48i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.372 - 0.645i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.735 - 1.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.48 + 4.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.66 - 8.08i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.78 + 10.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.49 + 6.05i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.09 - 5.35i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.95 + 5.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 + (-3.45 + 5.97i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.19 - 12.4i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.37506031867572064307202164034, −9.263227497034088987906925522225, −8.023010809440769214549608240831, −7.24505740606610509658572763600, −6.48921418854256480303960937867, −6.07178705598263308048850878476, −5.19770413577448695873670793695, −3.37815071265262314896191331859, −2.46476363596100134616944574226, −0.39997121756231062620752483480, 1.66270910217665653416022880891, 3.43217127087048107854052970296, 4.24248629997669673172384784304, 5.08204112346445261959214865456, 5.88284050570223244722653864957, 6.69951221855416929467086439785, 8.638438829102945844944621845962, 9.270155105128392909543051766288, 9.797119084395342671830662162828, 10.74033683425871646024864857415

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