↑

Properties

Label 2-3380-3380.2683-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.0876 + 0.996i$
Analytic cond. $1.68683$
Root an. cond. $1.29878$
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  + (0.663 − 0.748i)2-s + (−0.120 − 0.992i)4-s + (0.568 + 0.822i)5-s + (−0.822 − 0.568i)8-s + (0.663 − 0.748i)9-s + (0.992 + 0.120i)10-s + (−0.568 − 0.822i)13-s + (−0.970 + 0.239i)16-s + (1.21 − 0.222i)17-s + (−0.120 − 0.992i)18-s + (0.748 − 0.663i)20-s + (−0.354 + 0.935i)25-s + (−0.992 − 0.120i)26-s + (0.470 − 0.530i)29-s + (−0.464 + 0.885i)32-s + ⋯
L(s)  = 1  + (0.663 − 0.748i)2-s + (−0.120 − 0.992i)4-s + (0.568 + 0.822i)5-s + (−0.822 − 0.568i)8-s + (0.663 − 0.748i)9-s + (0.992 + 0.120i)10-s + (−0.568 − 0.822i)13-s + (−0.970 + 0.239i)16-s + (1.21 − 0.222i)17-s + (−0.120 − 0.992i)18-s + (0.748 − 0.663i)20-s + (−0.354 + 0.935i)25-s + (−0.992 − 0.120i)26-s + (0.470 − 0.530i)29-s + (−0.464 + 0.885i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0876 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0876 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.0876 + 0.996i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (2683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.0876 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.999310804\)
\(L(\frac12)\) \(\approx\) \(1.999310804\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.663 + 0.748i)T \)
5 \( 1 + (-0.568 - 0.822i)T \)
13 \( 1 + (0.568 + 0.822i)T \)
good3 \( 1 + (-0.663 + 0.748i)T^{2} \)
7 \( 1 + (0.354 + 0.935i)T^{2} \)
11 \( 1 + (-0.992 - 0.120i)T^{2} \)
17 \( 1 + (-1.21 + 0.222i)T + (0.935 - 0.354i)T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (-0.470 + 0.530i)T + (-0.120 - 0.992i)T^{2} \)
31 \( 1 + (0.822 + 0.568i)T^{2} \)
37 \( 1 + (0.423 - 0.222i)T + (0.568 - 0.822i)T^{2} \)
41 \( 1 + (-0.0495 - 0.110i)T + (-0.663 + 0.748i)T^{2} \)
43 \( 1 + (-0.822 + 0.568i)T^{2} \)
47 \( 1 + (0.970 + 0.239i)T^{2} \)
53 \( 1 + (-0.354 + 0.0649i)T + (0.935 - 0.354i)T^{2} \)
59 \( 1 + (-0.464 + 0.885i)T^{2} \)
61 \( 1 + (-1.06 - 1.53i)T + (-0.354 + 0.935i)T^{2} \)
67 \( 1 + (-0.970 - 0.239i)T^{2} \)
71 \( 1 + (0.663 - 0.748i)T^{2} \)
73 \( 1 + (1.32 + 1.49i)T + (-0.120 + 0.992i)T^{2} \)
79 \( 1 + (-0.970 - 0.239i)T^{2} \)
83 \( 1 + (0.748 - 0.663i)T^{2} \)
89 \( 1 + (-0.731 + 0.731i)T - iT^{2} \)
97 \( 1 + (-0.222 - 0.902i)T + (-0.885 + 0.464i)T^{2} \)
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

−8.893012928338024952151415791608, −7.68476606071090052660312250791, −6.98215265589979802728837791729, −6.20155211797680936929019402333, −5.56508439668734755194477476875, −4.76341772644745073546062801377, −3.67519558752047347163097596234, −3.11005783890541904973811872568, −2.22008966897877194807124565577, −1.05801875615589519658008164429, 1.54599263687048851251191325493, 2.56675567099713177059294925990, 3.77887068806797718454819584420, 4.60188748340269408846531433818, 5.13967750250662788533962366817, 5.82451514748038481380365552084, 6.72224694511929446998563412910, 7.41344167641285199553525683594, 8.112154744221184543311731958589, 8.787510180443510105599798954249

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