↑

Properties

Label 2-1456-13.4-c1-0-15
Degree $2$
Conductor $1456$
Sign $0.803 - 0.595i$
Analytic cond. $11.6262$
Root an. cond. $3.40972$
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  + (1.21 − 2.10i)3-s + 3.98i·5-s + (0.866 − 0.5i)7-s + (−1.45 − 2.52i)9-s + (−2.90 − 1.67i)11-s + (0.364 + 3.58i)13-s + (8.40 + 4.85i)15-s + (3.40 + 5.89i)17-s + (−0.621 + 0.359i)19-s − 2.43i·21-s + (−0.509 + 0.882i)23-s − 10.9·25-s + 0.200·27-s + (4.79 − 8.31i)29-s + 4.52i·31-s + ⋯
L(s)  = 1  + (0.702 − 1.21i)3-s + 1.78i·5-s + (0.327 − 0.188i)7-s + (−0.486 − 0.842i)9-s + (−0.875 − 0.505i)11-s + (0.100 + 0.994i)13-s + (2.16 + 1.25i)15-s + (0.825 + 1.43i)17-s + (−0.142 + 0.0823i)19-s − 0.530i·21-s + (−0.106 + 0.183i)23-s − 2.18·25-s + 0.0385·27-s + (0.891 − 1.54i)29-s + 0.812i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.803 - 0.595i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 0.803 - 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.038849322\)
\(L(\frac12)\) \(\approx\) \(2.038849322\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.364 - 3.58i)T \)
good3 \( 1 + (-1.21 + 2.10i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 3.98iT - 5T^{2} \)
11 \( 1 + (2.90 + 1.67i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.40 - 5.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.621 - 0.359i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.509 - 0.882i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.79 + 8.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.52iT - 31T^{2} \)
37 \( 1 + (-4.53 - 2.61i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.985 + 0.568i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.25 - 7.37i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 11.0iT - 47T^{2} \)
53 \( 1 - 3.19T + 53T^{2} \)
59 \( 1 + (6.50 - 3.75i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.90 - 3.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.6 + 7.30i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.0 + 6.39i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.44iT - 73T^{2} \)
79 \( 1 + 3.61T + 79T^{2} \)
83 \( 1 + 3.31iT - 83T^{2} \)
89 \( 1 + (-5.96 - 3.44i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.3 - 5.97i)T + (48.5 - 84.0i)T^{2} \)
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

−9.677446088794207309502702894707, −8.388348101506273817362487413682, −7.85845469632273388715174455382, −7.33636047243672231677298690818, −6.35989195915948677234504000699, −6.04262856083971983486330705388, −4.28881880085622846954282787014, −3.15604534624045657990513836955, −2.52892283010375205112455643073, −1.51294330562379350557915099693, 0.77791887200849392320895218395, 2.40927949571220831635296276142, 3.48555159802540316373015858894, 4.55890212555967475119206993581, 5.07068762514784007943866054433, 5.58015930771681756404279427488, 7.37762585179102809001562900756, 8.155445137288145122510019657435, 8.728772026503517444487601641019, 9.370017891308690933670294357894

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