Properties

Label 2-675-1.1-c3-0-47
Degree $2$
Conductor $675$
Sign $-1$
Analytic cond. $39.8262$
Root an. cond. $6.31080$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.258·2-s − 7.93·4-s − 14.5·7-s + 4.12·8-s + 49.2·11-s − 72.1·13-s + 3.75·14-s + 62.3·16-s + 118.·17-s + 123.·19-s − 12.7·22-s − 91.4·23-s + 18.6·26-s + 115.·28-s − 174.·29-s − 46.2·31-s − 49.1·32-s − 30.5·34-s − 154.·37-s − 31.9·38-s + 364.·41-s − 125.·43-s − 390.·44-s + 23.6·46-s − 221.·47-s − 132.·49-s + 572.·52-s + ⋯
L(s)  = 1  − 0.0914·2-s − 0.991·4-s − 0.783·7-s + 0.182·8-s + 1.35·11-s − 1.53·13-s + 0.0716·14-s + 0.974·16-s + 1.68·17-s + 1.48·19-s − 0.123·22-s − 0.829·23-s + 0.140·26-s + 0.777·28-s − 1.11·29-s − 0.268·31-s − 0.271·32-s − 0.154·34-s − 0.688·37-s − 0.136·38-s + 1.38·41-s − 0.445·43-s − 1.33·44-s + 0.0758·46-s − 0.687·47-s − 0.385·49-s + 1.52·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(39.8262\)
Root analytic conductor: \(6.31080\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 675,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2 \( 1 + 0.258T + 8T^{2} \)
7 \( 1 + 14.5T + 343T^{2} \)
11 \( 1 - 49.2T + 1.33e3T^{2} \)
13 \( 1 + 72.1T + 2.19e3T^{2} \)
17 \( 1 - 118.T + 4.91e3T^{2} \)
19 \( 1 - 123.T + 6.85e3T^{2} \)
23 \( 1 + 91.4T + 1.21e4T^{2} \)
29 \( 1 + 174.T + 2.43e4T^{2} \)
31 \( 1 + 46.2T + 2.97e4T^{2} \)
37 \( 1 + 154.T + 5.06e4T^{2} \)
41 \( 1 - 364.T + 6.89e4T^{2} \)
43 \( 1 + 125.T + 7.95e4T^{2} \)
47 \( 1 + 221.T + 1.03e5T^{2} \)
53 \( 1 + 13.6T + 1.48e5T^{2} \)
59 \( 1 + 239.T + 2.05e5T^{2} \)
61 \( 1 + 54.5T + 2.26e5T^{2} \)
67 \( 1 - 76.0T + 3.00e5T^{2} \)
71 \( 1 + 728.T + 3.57e5T^{2} \)
73 \( 1 - 501.T + 3.89e5T^{2} \)
79 \( 1 - 397.T + 4.93e5T^{2} \)
83 \( 1 + 1.36e3T + 5.71e5T^{2} \)
89 \( 1 + 1.46e3T + 7.04e5T^{2} \)
97 \( 1 + 335.T + 9.12e5T^{2} \)
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

−9.735319852133442835443044694697, −9.106309948557660823742550032545, −7.84571209656769414474315739518, −7.20308636640991476464467929224, −5.91313362681760704864163283228, −5.12452559077187558901512581917, −3.94257719189974779877219500002, −3.15231960446786648172493864248, −1.32324874297173100205937929991, 0, 1.32324874297173100205937929991, 3.15231960446786648172493864248, 3.94257719189974779877219500002, 5.12452559077187558901512581917, 5.91313362681760704864163283228, 7.20308636640991476464467929224, 7.84571209656769414474315739518, 9.106309948557660823742550032545, 9.735319852133442835443044694697

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