↑

Properties

Label 2-1568-28.19-c1-0-15
Degree $2$
Conductor $1568$
Sign $0.995 + 0.0932i$
Analytic cond. $12.5205$
Root an. cond. $3.53843$
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  + (−0.826 − 1.43i)3-s + (3.03 + 1.75i)5-s + (0.132 − 0.230i)9-s + (−2.64 + 1.52i)11-s + 3.22i·13-s − 5.78i·15-s + (−1.36 + 0.786i)17-s + (4.03 − 6.99i)19-s + (7.20 + 4.16i)23-s + (3.62 + 6.28i)25-s − 5.40·27-s + 8.17·29-s + (−1.53 − 2.65i)31-s + (4.37 + 2.52i)33-s + (−1.17 + 2.04i)37-s + ⋯
L(s)  = 1  + (−0.477 − 0.826i)3-s + (1.35 + 0.782i)5-s + (0.0442 − 0.0766i)9-s + (−0.797 + 0.460i)11-s + 0.894i·13-s − 1.49i·15-s + (−0.330 + 0.190i)17-s + (0.926 − 1.60i)19-s + (1.50 + 0.867i)23-s + (0.725 + 1.25i)25-s − 1.03·27-s + 1.51·29-s + (−0.274 − 0.476i)31-s + (0.761 + 0.439i)33-s + (−0.193 + 0.335i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.995 + 0.0932i$
Analytic conductor: \(12.5205\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ 0.995 + 0.0932i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.878847972\)
\(L(\frac12)\) \(\approx\) \(1.878847972\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 + (0.826 + 1.43i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-3.03 - 1.75i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.64 - 1.52i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.22iT - 13T^{2} \)
17 \( 1 + (1.36 - 0.786i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.03 + 6.99i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.20 - 4.16i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.17T + 29T^{2} \)
31 \( 1 + (1.53 + 2.65i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.17 - 2.04i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.24iT - 41T^{2} \)
43 \( 1 - 8.71iT - 43T^{2} \)
47 \( 1 + (1.65 - 2.86i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.93 - 6.81i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.35 - 5.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.19 - 5.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.48 + 4.32i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.42iT - 71T^{2} \)
73 \( 1 + (1.31 - 0.756i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.26 + 1.88i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.13T + 83T^{2} \)
89 \( 1 + (7.71 + 4.45i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.85iT - 97T^{2} \)
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

−9.509157760030907565258858369245, −8.802608323579435615337118665395, −7.29064075460885171231587242085, −7.05583871653631135017596036163, −6.30725318086050786490185235784, −5.49646669188575309559826768382, −4.64954981066003458297171031759, −3.01538923613880772145871005545, −2.24200217893474127284266838978, −1.13589181887850363612068252812, 0.958415453883563196821658970457, 2.32151889573058984380029852599, 3.46933872374624764825389965757, 4.84730319701795987005151026699, 5.26398964794750813782703870755, 5.79338171233416526394229254396, 6.87694184798230768633091090286, 8.127659995825793781589780068395, 8.687136619413725747211155931979, 9.735405250633049332005081180505

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