↑

Properties

Label 2-1040-13.4-c1-0-18
Degree $2$
Conductor $1040$
Sign $0.462 + 0.886i$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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  + (−1.43 + 2.48i)3-s + i·5-s + (−0.0113 + 0.00658i)7-s + (−2.62 − 4.53i)9-s + (−0.541 − 0.312i)11-s + (1.97 − 3.01i)13-s + (−2.48 − 1.43i)15-s + (−3.78 − 6.54i)17-s + (−6.29 + 3.63i)19-s − 0.0377i·21-s + (1.22 − 2.13i)23-s − 25-s + 6.43·27-s + (−1.15 + 2.00i)29-s − 8.02i·31-s + ⋯
L(s)  = 1  + (−0.828 + 1.43i)3-s + 0.447i·5-s + (−0.00430 + 0.00248i)7-s + (−0.873 − 1.51i)9-s + (−0.163 − 0.0942i)11-s + (0.547 − 0.837i)13-s + (−0.641 − 0.370i)15-s + (−0.916 − 1.58i)17-s + (−1.44 + 0.833i)19-s − 0.00824i·21-s + (0.256 − 0.444i)23-s − 0.200·25-s + 1.23·27-s + (−0.215 + 0.373i)29-s − 1.44i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.462 + 0.886i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.462 + 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4070866200\)
\(L(\frac12)\) \(\approx\) \(0.4070866200\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (-1.97 + 3.01i)T \)
good3 \( 1 + (1.43 - 2.48i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.0113 - 0.00658i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.541 + 0.312i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.78 + 6.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.29 - 3.63i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.22 + 2.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.15 - 2.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.02iT - 31T^{2} \)
37 \( 1 + (3.65 + 2.10i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.29 - 1.90i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.39 + 2.42i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.00iT - 47T^{2} \)
53 \( 1 - 6.25T + 53T^{2} \)
59 \( 1 + (-11.6 + 6.74i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.28 - 5.69i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.81 + 3.35i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (13.2 - 7.63i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.98iT - 73T^{2} \)
79 \( 1 + 2.02T + 79T^{2} \)
83 \( 1 - 2.35iT - 83T^{2} \)
89 \( 1 + (-4.50 - 2.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.02 + 4.05i)T + (48.5 - 84.0i)T^{2} \)
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

−9.998958040459407738871100592952, −9.156077828193233518142350818569, −8.337280807622230600791492385384, −7.10268350826552109255177792476, −6.12752016504626232114828091140, −5.44481766617991869286016287280, −4.49273994636149192884668715749, −3.74066546835560461583392232826, −2.56592253154396949762769617525, −0.20682438651421476396378786095, 1.38705145979672824122522194713, 2.20988289246423149985550880488, 4.00860611454377841700177406674, 5.02717569432082410802209949652, 6.12757448544047983155171865086, 6.57652549903000703632979542412, 7.36372728536924876529680790591, 8.512885089763541393434944093787, 8.816020769295204949985264964784, 10.34999472645599197665059987022

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