↑

Properties

Label 2-2475-165.32-c0-0-0
Degree $2$
Conductor $2475$
Sign $-0.391 - 0.920i$
Analytic cond. $1.23518$
Root an. cond. $1.11138$
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  + (−1.30 − 1.30i)2-s + 2.41i·4-s + (−1.30 + 1.30i)7-s + (1.84 − 1.84i)8-s + i·11-s + (0.541 + 0.541i)13-s + 3.41·14-s − 2.41·16-s + (−0.541 − 0.541i)17-s + (1.30 − 1.30i)22-s − 1.41i·26-s + (−3.15 − 3.15i)28-s − 1.41·31-s + (1.30 + 1.30i)32-s + 1.41i·34-s + ⋯
L(s)  = 1  + (−1.30 − 1.30i)2-s + 2.41i·4-s + (−1.30 + 1.30i)7-s + (1.84 − 1.84i)8-s + i·11-s + (0.541 + 0.541i)13-s + 3.41·14-s − 2.41·16-s + (−0.541 − 0.541i)17-s + (1.30 − 1.30i)22-s − 1.41i·26-s + (−3.15 − 3.15i)28-s − 1.41·31-s + (1.30 + 1.30i)32-s + 1.41i·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.391 - 0.920i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1682, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ -0.391 - 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1605152935\)
\(L(\frac12)\) \(\approx\) \(0.1605152935\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 - iT \)
good2 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
7 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
13 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
17 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + iT^{2} \)
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

−9.431766865619911978305047408910, −8.959357478156911925027559609236, −8.254416592948524790190334680702, −7.18420053354601923507196134970, −6.60297541780289150879776716267, −5.45771115350467459626405929821, −4.16239652320268263522846368226, −3.26639541270493178752229877931, −2.47414807903281301142891732027, −1.73122396721023168590512940690, 0.17172685341429072668836354808, 1.33585774044549477518060623040, 3.17716774488031901296832119395, 4.09795156210934012381078573058, 5.42852886125833809039159881527, 6.18339848934126575962119831397, 6.63662269024704373829359134612, 7.39575276900780403385881592529, 8.050855236128467750538914036056, 8.813532379536337593945674436204

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