↑

Properties

Label 2-3381-483.482-c0-0-3
Degree $2$
Conductor $3381$
Sign $0.999 - 0.0287i$
Analytic cond. $1.68733$
Root an. cond. $1.29897$
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  − 1.41i·2-s + (−0.382 + 0.923i)3-s − 1.00·4-s + (1.30 + 0.541i)6-s + (−0.707 − 0.707i)9-s + (0.382 − 0.923i)12-s + 1.84i·13-s − 0.999·16-s + (−1.00 + i)18-s + i·23-s + 25-s + 2.61·26-s + (0.923 − 0.382i)27-s + 0.765i·31-s + 1.41i·32-s + ⋯
L(s)  = 1  − 1.41i·2-s + (−0.382 + 0.923i)3-s − 1.00·4-s + (1.30 + 0.541i)6-s + (−0.707 − 0.707i)9-s + (0.382 − 0.923i)12-s + 1.84i·13-s − 0.999·16-s + (−1.00 + i)18-s + i·23-s + 25-s + 2.61·26-s + (0.923 − 0.382i)27-s + 0.765i·31-s + 1.41i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.999 - 0.0287i$
Analytic conductor: \(1.68733\)
Root analytic conductor: \(1.29897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (3380, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3381,\ (\ :0),\ 0.999 - 0.0287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9342115505\)
\(L(\frac12)\) \(\approx\) \(0.9342115505\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 \)
23 \( 1 - iT \)
good2 \( 1 + 1.41iT - T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.84iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 0.765iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.84T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.84T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 0.765T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 - 0.765iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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

−8.978178521809915564759604303671, −8.677485078567921673261771334471, −7.10835218672521096270674286604, −6.59724147949785604424052250082, −5.45845353611379567815614716999, −4.65304284269505918677081053088, −3.99105414056150104379449054721, −3.33096893665312897667316842884, −2.33053637142508769041113624568, −1.26869941500049085455858799449, 0.63209559677223529835477854573, 2.22496897380827036615408983890, 3.17900594383252940111808463956, 4.66226507136307246866979167211, 5.33922091881888158340309043705, 5.92437800751161018234809141369, 6.59138181807778821410236015072, 7.26080289217988716180095190753, 7.894204254689163156317384905178, 8.387346674922328009239947154983

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