Properties

Label 2-1875-1.1-c1-0-64
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  + 2.68·2-s + 3-s + 5.22·4-s + 2.68·6-s − 1.68·7-s + 8.65·8-s + 9-s + 1.07·11-s + 5.22·12-s − 2.67·13-s − 4.53·14-s + 12.8·16-s + 3.93·17-s + 2.68·18-s − 1.17·19-s − 1.68·21-s + 2.89·22-s + 4.06·23-s + 8.65·24-s − 7.19·26-s + 27-s − 8.80·28-s + 5.95·29-s − 7.10·31-s + 17.1·32-s + 1.07·33-s + 10.5·34-s + ⋯
L(s)  = 1  + 1.90·2-s + 0.577·3-s + 2.61·4-s + 1.09·6-s − 0.637·7-s + 3.05·8-s + 0.333·9-s + 0.325·11-s + 1.50·12-s − 0.742·13-s − 1.21·14-s + 3.20·16-s + 0.953·17-s + 0.633·18-s − 0.270·19-s − 0.368·21-s + 0.618·22-s + 0.847·23-s + 1.76·24-s − 1.41·26-s + 0.192·27-s − 1.66·28-s + 1.10·29-s − 1.27·31-s + 3.02·32-s + 0.187·33-s + 1.81·34-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

\(L(1)\) \(\approx\) \(6.858624900\)
\(L(\frac12)\) \(\approx\) \(6.858624900\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.68T + 2T^{2} \)
7 \( 1 + 1.68T + 7T^{2} \)
11 \( 1 - 1.07T + 11T^{2} \)
13 \( 1 + 2.67T + 13T^{2} \)
17 \( 1 - 3.93T + 17T^{2} \)
19 \( 1 + 1.17T + 19T^{2} \)
23 \( 1 - 4.06T + 23T^{2} \)
29 \( 1 - 5.95T + 29T^{2} \)
31 \( 1 + 7.10T + 31T^{2} \)
37 \( 1 - 4.58T + 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + 2.58T + 43T^{2} \)
47 \( 1 - 1.91T + 47T^{2} \)
53 \( 1 - 2.54T + 53T^{2} \)
59 \( 1 - 1.33T + 59T^{2} \)
61 \( 1 + 7.28T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 + 5.98T + 71T^{2} \)
73 \( 1 - 3.30T + 73T^{2} \)
79 \( 1 + 4.00T + 79T^{2} \)
83 \( 1 + 8.87T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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

−9.347148158103630459854471321508, −8.189417486942495867357847182576, −7.24232833872217393075359799335, −6.76010148091907997322553037167, −5.86065579386260884411186778493, −5.04264147000754249727750399042, −4.25756396562676213225006146401, −3.30596872311980354847481721425, −2.83797646395518185128656680628, −1.63601948830340604688332862160, 1.63601948830340604688332862160, 2.83797646395518185128656680628, 3.30596872311980354847481721425, 4.25756396562676213225006146401, 5.04264147000754249727750399042, 5.86065579386260884411186778493, 6.76010148091907997322553037167, 7.24232833872217393075359799335, 8.189417486942495867357847182576, 9.347148158103630459854471321508

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