↑

Properties

Label 2-1352-104.75-c0-0-6
Degree $2$
Conductor $1352$
Sign $0.669 + 0.743i$
Analytic cond. $0.674735$
Root an. cond. $0.821423$
Motivic weight $0$
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)2-s + (0.222 + 0.385i)3-s + (0.499 − 0.866i)4-s + (0.385 + 0.222i)6-s − 0.999i·8-s + (0.400 − 0.694i)9-s + (−1.07 + 0.623i)11-s + 0.445·12-s + (−0.5 − 0.866i)16-s + (0.623 − 1.07i)17-s − 0.801i·18-s + (1.56 + 0.900i)19-s + (−0.623 + 1.07i)22-s + (0.385 − 0.222i)24-s − 25-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.222 + 0.385i)3-s + (0.499 − 0.866i)4-s + (0.385 + 0.222i)6-s − 0.999i·8-s + (0.400 − 0.694i)9-s + (−1.07 + 0.623i)11-s + 0.445·12-s + (−0.5 − 0.866i)16-s + (0.623 − 1.07i)17-s − 0.801i·18-s + (1.56 + 0.900i)19-s + (−0.623 + 1.07i)22-s + (0.385 − 0.222i)24-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $0.669 + 0.743i$
Analytic conductor: \(0.674735\)
Root analytic conductor: \(0.821423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :0),\ 0.669 + 0.743i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.885434527\)
\(L(\frac12)\) \(\approx\) \(1.885434527\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 \)
good3 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (1.07 - 0.623i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1.56 - 0.900i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (1.56 - 0.900i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.385 + 0.222i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.385 - 0.222i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - 0.445iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.80iT - T^{2} \)
89 \( 1 + (-0.385 + 0.222i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.07 + 0.623i)T + (0.5 + 0.866i)T^{2} \)
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

−9.947573704471808424804716838037, −9.315300174221624069830865506490, −7.87171851332043315986113897498, −7.24531904435347906534259870808, −6.17483118287280791308318139474, −5.25802507390676725488262451037, −4.61502453000986182777598518409, −3.49037766199102725360602020451, −2.86913351189514961967768555911, −1.43391724233009600052326448613, 1.90080608051119407399955955325, 2.98704715915442080982626164701, 3.86019661437208942056310414750, 5.20008772859537051104516041876, 5.46925186723726641914916442036, 6.69152068034646531773159315800, 7.47169952248103546476249707989, 8.023027117622335859611853840620, 8.730941550457539315065709006956, 10.07555760725864644340313850001

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