Properties

Label 2-1875-1.1-c1-0-57
Degree $2$
Conductor $1875$
Sign $1$
Analytic cond. $14.9719$
Root an. cond. $3.86936$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.69·2-s − 3-s + 5.27·4-s − 2.69·6-s + 3.56·7-s + 8.84·8-s + 9-s − 0.0695·11-s − 5.27·12-s − 4.85·13-s + 9.62·14-s + 13.2·16-s − 3.03·17-s + 2.69·18-s + 5.05·19-s − 3.56·21-s − 0.187·22-s + 7.43·23-s − 8.84·24-s − 13.1·26-s − 27-s + 18.8·28-s + 1.95·29-s − 5.63·31-s + 18.1·32-s + 0.0695·33-s − 8.18·34-s + ⋯
L(s)  = 1  + 1.90·2-s − 0.577·3-s + 2.63·4-s − 1.10·6-s + 1.34·7-s + 3.12·8-s + 0.333·9-s − 0.0209·11-s − 1.52·12-s − 1.34·13-s + 2.57·14-s + 3.32·16-s − 0.735·17-s + 0.635·18-s + 1.15·19-s − 0.778·21-s − 0.0400·22-s + 1.54·23-s − 1.80·24-s − 2.56·26-s − 0.192·27-s + 3.55·28-s + 0.362·29-s − 1.01·31-s + 3.21·32-s + 0.0121·33-s − 1.40·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(14.9719\)
Root analytic conductor: \(3.86936\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.627266978\)
\(L(\frac12)\) \(\approx\) \(5.627266978\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
good2 \( 1 - 2.69T + 2T^{2} \)
7 \( 1 - 3.56T + 7T^{2} \)
11 \( 1 + 0.0695T + 11T^{2} \)
13 \( 1 + 4.85T + 13T^{2} \)
17 \( 1 + 3.03T + 17T^{2} \)
19 \( 1 - 5.05T + 19T^{2} \)
23 \( 1 - 7.43T + 23T^{2} \)
29 \( 1 - 1.95T + 29T^{2} \)
31 \( 1 + 5.63T + 31T^{2} \)
37 \( 1 + 6.21T + 37T^{2} \)
41 \( 1 + 5.63T + 41T^{2} \)
43 \( 1 + 0.244T + 43T^{2} \)
47 \( 1 - 3.23T + 47T^{2} \)
53 \( 1 + 8.37T + 53T^{2} \)
59 \( 1 + 1.60T + 59T^{2} \)
61 \( 1 - 3.12T + 61T^{2} \)
67 \( 1 - 2.94T + 67T^{2} \)
71 \( 1 - 7.25T + 71T^{2} \)
73 \( 1 + 3.69T + 73T^{2} \)
79 \( 1 - 4.70T + 79T^{2} \)
83 \( 1 - 8.80T + 83T^{2} \)
89 \( 1 + 3.55T + 89T^{2} \)
97 \( 1 + 8.80T + 97T^{2} \)
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

−9.348731714254963175778186657765, −8.026677696913516587065328193608, −7.19560953880284008282653765235, −6.80293972975355879667336970765, −5.56956053370863526685052426451, −4.98083410453394385839500661443, −4.72921593852825827925941934472, −3.55540253097291343130710326716, −2.49258455892223997887128397071, −1.50686460071371744456422445347, 1.50686460071371744456422445347, 2.49258455892223997887128397071, 3.55540253097291343130710326716, 4.72921593852825827925941934472, 4.98083410453394385839500661443, 5.56956053370863526685052426451, 6.80293972975355879667336970765, 7.19560953880284008282653765235, 8.026677696913516587065328193608, 9.348731714254963175778186657765

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