↑

Properties

Label 2-1520-19.11-c1-0-32
Degree $2$
Conductor $1520$
Sign $-0.0977 + 0.995i$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
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  + (1.14 − 1.97i)3-s + (0.5 − 0.866i)5-s + 1.28·7-s + (−1.11 − 1.92i)9-s − 0.285·11-s + (2.5 + 4.33i)13-s + (−1.14 − 1.97i)15-s + (3.11 − 5.40i)17-s + (−2.92 − 3.22i)19-s + (1.46 − 2.54i)21-s + (−2.61 − 4.53i)23-s + (−0.499 − 0.866i)25-s + 1.77·27-s + (−0.642 − 1.11i)29-s + 1.22·31-s + ⋯
L(s)  = 1  + (0.659 − 1.14i)3-s + (0.223 − 0.387i)5-s + 0.485·7-s + (−0.370 − 0.641i)9-s − 0.0859·11-s + (0.693 + 1.20i)13-s + (−0.295 − 0.510i)15-s + (0.756 − 1.30i)17-s + (−0.671 − 0.740i)19-s + (0.320 − 0.554i)21-s + (−0.545 − 0.945i)23-s + (−0.0999 − 0.173i)25-s + 0.342·27-s + (−0.119 − 0.206i)29-s + 0.219·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.0977 + 0.995i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ -0.0977 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.389751486\)
\(L(\frac12)\) \(\approx\) \(2.389751486\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (2.92 + 3.22i)T \)
good3 \( 1 + (-1.14 + 1.97i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 1.28T + 7T^{2} \)
11 \( 1 + 0.285T + 11T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.11 + 5.40i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.61 + 4.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.642 + 1.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.22T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 + (-0.420 + 0.728i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.47 - 4.28i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.86 - 4.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.18 + 10.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.86 + 4.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.22 + 3.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.492 + 0.853i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.46 - 2.53i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.382 + 0.661i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.72 - 13.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.66T + 83T^{2} \)
89 \( 1 + (-8.01 - 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.87 - 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.144846387231404695953123818424, −8.249253566701984935374953433446, −7.85581279671655334645327901202, −6.80413714993565585429501018599, −6.31610372907602064507968593504, −5.02946612469619260493867962738, −4.22879325221765461172843269847, −2.79829826178496471825122020489, −1.99167756695636966760785121610, −0.944318733098938871439714890422, 1.56318954950286454183937721118, 2.95617078311195256252836179522, 3.68091204074545191807288629790, 4.42340920034426846235560294137, 5.64002946708075372932532867667, 6.12358946264852160603495150059, 7.63040332608541604585946957023, 8.166763270534069935490941431280, 8.871576812362627070543139048659, 9.839639460126872636387437896524

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