↑

Properties

Label 2-2565-2565.94-c0-0-5
Degree $2$
Conductor $2565$
Sign $-0.116 + 0.993i$
Analytic cond. $1.28010$
Root an. cond. $1.13141$
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  + (−0.266 − 1.50i)2-s + (−0.766 + 0.642i)3-s + (−1.26 + 0.460i)4-s + (0.766 + 0.642i)5-s + (1.17 + 0.984i)6-s + (0.266 + 0.460i)8-s + (0.173 − 0.984i)9-s + (0.766 − 1.32i)10-s + (1.17 − 0.984i)11-s + (0.673 − 1.16i)12-s + (−0.0603 + 0.342i)13-s − 15-s + (−0.407 + 0.342i)16-s − 1.53·18-s + (−0.5 − 0.866i)19-s + (−1.26 − 0.460i)20-s + ⋯
L(s)  = 1  + (−0.266 − 1.50i)2-s + (−0.766 + 0.642i)3-s + (−1.26 + 0.460i)4-s + (0.766 + 0.642i)5-s + (1.17 + 0.984i)6-s + (0.266 + 0.460i)8-s + (0.173 − 0.984i)9-s + (0.766 − 1.32i)10-s + (1.17 − 0.984i)11-s + (0.673 − 1.16i)12-s + (−0.0603 + 0.342i)13-s − 15-s + (−0.407 + 0.342i)16-s − 1.53·18-s + (−0.5 − 0.866i)19-s + (−1.26 − 0.460i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2565 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2565 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2565\)    =    \(3^{3} \cdot 5 \cdot 19\)
Sign: $-0.116 + 0.993i$
Analytic conductor: \(1.28010\)
Root analytic conductor: \(1.13141\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2565} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2565,\ (\ :0),\ -0.116 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9136503984\)
\(L(\frac12)\) \(\approx\) \(0.9136503984\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
7 \( 1 + (-0.766 - 0.642i)T^{2} \)
11 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
13 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (-0.766 + 0.642i)T^{2} \)
37 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 - 1.87T + T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.939 - 0.342i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
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

−9.155180704075687283756526022742, −8.803407110155593276687196101315, −7.16682941270589145184687649110, −6.34588990924487249019823768974, −5.83306208672018981494743388880, −4.61694069704153168904344986290, −3.84324991853515416878801298116, −3.10055416580226036281160401157, −2.07617467934664556333921500763, −0.876926877311168072125633145372, 1.19260296659901377364116260491, 2.26567858904092628826844105133, 4.24286661724631040312752765971, 4.92091225130166438184970767698, 5.75634726445344372569230597703, 6.22759494204969537662069262086, 6.89357210871233638729674721479, 7.56130686923240675079303540123, 8.374007580225888826439125490530, 8.994500018230691250273204065508

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