↑

Properties

Label 2-1568-28.19-c1-0-1
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  + (0.382 + 0.662i)3-s + (−0.937 − 0.541i)5-s + (1.20 − 2.09i)9-s + (−3.67 + 2.12i)11-s + 2.61i·13-s − 0.828i·15-s + (−1.05 + 0.606i)17-s + (−1.68 + 2.92i)19-s + (−3.16 − 1.82i)23-s + (−1.91 − 3.31i)25-s + 4.14·27-s − 4·29-s + (−3.69 − 6.40i)31-s + (−2.81 − 1.62i)33-s + (2.58 − 4.47i)37-s + ⋯
L(s)  = 1  + (0.220 + 0.382i)3-s + (−0.419 − 0.242i)5-s + (0.402 − 0.696i)9-s + (−1.10 + 0.639i)11-s + 0.724i·13-s − 0.213i·15-s + (−0.254 + 0.147i)17-s + (−0.387 + 0.671i)19-s + (−0.660 − 0.381i)23-s + (−0.382 − 0.663i)25-s + 0.797·27-s − 0.742·29-s + (−0.663 − 1.14i)31-s + (−0.489 − 0.282i)33-s + (0.425 − 0.736i)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

\(L(1)\) \(\approx\) \(0.2565550076\)
\(L(\frac12)\) \(\approx\) \(0.2565550076\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.382 - 0.662i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.937 + 0.541i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.67 - 2.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.61iT - 13T^{2} \)
17 \( 1 + (1.05 - 0.606i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.68 - 2.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.16 + 1.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (3.69 + 6.40i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.58 + 4.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.68iT - 41T^{2} \)
43 \( 1 - 6.58iT - 43T^{2} \)
47 \( 1 + (6.62 - 11.4i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.58 + 2.74i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.91 - 11.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (13.1 + 7.61i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.36 - 4.82i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.65iT - 71T^{2} \)
73 \( 1 + (-3.86 + 2.23i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.37 + 5.41i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.48T + 83T^{2} \)
89 \( 1 + (8.38 + 4.84i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.01iT - 97T^{2} \)
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

−9.701393382116553378960972958680, −9.202770988034085984342397903462, −8.079692147890016694496226913978, −7.67121211283300271263192206072, −6.55068518826379490920678594492, −5.79473595784396468084988881089, −4.37138645328999211637352114170, −4.27698478632164356187059322982, −2.92777081377832751450435091956, −1.75352189978894634719143953814, 0.090486616547884252882457226533, 1.84143719519460001877159741512, 2.87151117266749278077041119430, 3.78299561860105847701069153975, 5.04298610250047538323409787794, 5.61157349919691126187148334687, 6.88646727376814117814220189828, 7.49027267899372311010314187901, 8.153836938210727215197390367742, 8.822281440151911765816440292174

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