↑

Properties

Label 2-1520-19.7-c1-0-26
Degree $2$
Conductor $1520$
Sign $-0.813 + 0.582i$
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  + (−1.25 − 2.17i)3-s + (0.5 + 0.866i)5-s − 3.50·7-s + (−1.64 + 2.84i)9-s + 4.50·11-s + (2.5 − 4.33i)13-s + (1.25 − 2.17i)15-s + (−0.0793 − 0.137i)17-s + (4.26 + 0.920i)19-s + (4.39 + 7.61i)21-s + (0.579 − 1.00i)23-s + (−0.499 + 0.866i)25-s + 0.714·27-s + (1.75 − 3.03i)29-s + 2.28·31-s + ⋯
L(s)  = 1  + (−0.723 − 1.25i)3-s + (0.223 + 0.387i)5-s − 1.32·7-s + (−0.547 + 0.948i)9-s + 1.35·11-s + (0.693 − 1.20i)13-s + (0.323 − 0.560i)15-s + (−0.0192 − 0.0333i)17-s + (0.977 + 0.211i)19-s + (0.959 + 1.66i)21-s + (0.120 − 0.209i)23-s + (−0.0999 + 0.173i)25-s + 0.137·27-s + (0.325 − 0.563i)29-s + 0.410·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.813 + 0.582i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ -0.813 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9804481828\)
\(L(\frac12)\) \(\approx\) \(0.9804481828\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.26 - 0.920i)T \)
good3 \( 1 + (1.25 + 2.17i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 3.50T + 7T^{2} \)
11 \( 1 - 4.50T + 11T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.0793 + 0.137i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.579 + 1.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.75 + 3.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.28T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 + (3.03 + 5.26i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.67 + 2.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.53 + 2.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.87 + 4.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.53 - 2.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.436 + 0.756i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.22 - 7.31i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.11 + 14.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.57 - 6.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.06 + 8.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.85T + 83T^{2} \)
89 \( 1 + (-0.556 + 0.963i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.809 + 1.40i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.151537348616897590367693753105, −8.248735832146240330498200948559, −7.11914361412449133504776725376, −6.77972013175933117395599969826, −6.02395282628553196293622365638, −5.47188080743017124180749515629, −3.77475849069243258570198178211, −3.01256951842232508180883009105, −1.57928678146129614105410370352, −0.47292153138323557865301189351, 1.34408463933441205796984292032, 3.23177797785476918331971850757, 3.92378757235848158182085326267, 4.70997868820301282381402693261, 5.67705162834415201386903862316, 6.44002014884025505809570914510, 7.00354351708473399298842234315, 8.654102068815805166671344738586, 9.260006964952486706698050782273, 9.682588872240949443337672855136

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