↑

Properties

Label 2-1560-65.4-c1-0-12
Degree $2$
Conductor $1560$
Sign $0.0762 - 0.997i$
Analytic cond. $12.4566$
Root an. cond. $3.52939$
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 + (1.63 + 1.52i)5-s + (1.13 + 1.96i)7-s + (0.499 + 0.866i)9-s + (2.47 + 1.43i)11-s + (−1.36 + 3.33i)13-s + (−0.656 − 2.13i)15-s + (6.33 − 3.65i)17-s + (−6.64 + 3.83i)19-s − 2.26i·21-s + (−0.617 − 0.356i)23-s + (0.361 + 4.98i)25-s − 0.999i·27-s + (−3.35 + 5.81i)29-s − 2.56i·31-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (0.732 + 0.681i)5-s + (0.428 + 0.741i)7-s + (0.166 + 0.288i)9-s + (0.746 + 0.431i)11-s + (−0.377 + 0.925i)13-s + (−0.169 − 0.551i)15-s + (1.53 − 0.886i)17-s + (−1.52 + 0.879i)19-s − 0.494i·21-s + (−0.128 − 0.0743i)23-s + (0.0722 + 0.997i)25-s − 0.192i·27-s + (−0.623 + 1.08i)29-s − 0.460i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0762 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0762 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.0762 - 0.997i$
Analytic conductor: \(12.4566\)
Root analytic conductor: \(3.52939\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :1/2),\ 0.0762 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.597459166\)
\(L(\frac12)\) \(\approx\) \(1.597459166\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-1.63 - 1.52i)T \)
13 \( 1 + (1.36 - 3.33i)T \)
good7 \( 1 + (-1.13 - 1.96i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.47 - 1.43i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-6.33 + 3.65i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.64 - 3.83i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.617 + 0.356i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.35 - 5.81i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.56iT - 31T^{2} \)
37 \( 1 + (-3.00 + 5.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.15 + 2.40i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.62 + 0.935i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.49T + 47T^{2} \)
53 \( 1 - 8.14iT - 53T^{2} \)
59 \( 1 + (8.96 - 5.17i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.359 - 0.623i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.93 + 5.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.08 - 3.51i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + 9.81T + 79T^{2} \)
83 \( 1 - 17.6T + 83T^{2} \)
89 \( 1 + (-13.0 - 7.54i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.88 - 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.570031505103962656136353762228, −9.047962967974907810942870738101, −7.88399793045223680853976833339, −7.09336857397194147897216056137, −6.34463723972029395764238163356, −5.66450428944345070437018914592, −4.82339023706590900806041150734, −3.64237213018143991608891716608, −2.30332522785117906744601980255, −1.59576748733528290194324537838, 0.68291933655768288385856496346, 1.76723598722929273002961171687, 3.33482069849346140618699440781, 4.36002713605096466350371286136, 5.07376957364194929054671514465, 5.98393034488636227274727648940, 6.55828660462723224091530516300, 7.85039599957201907905799587572, 8.376413337100977733656473647483, 9.410645035816230091379107171477

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