Properties

Label 2-43e2-1.1-c1-0-70
Degree $2$
Conductor $1849$
Sign $-1$
Analytic cond. $14.7643$
Root an. cond. $3.84243$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.81·2-s − 1.25·3-s + 1.30·4-s − 0.160·5-s + 2.28·6-s + 4.31·7-s + 1.25·8-s − 1.42·9-s + 0.291·10-s − 0.148·11-s − 1.64·12-s + 1.68·13-s − 7.85·14-s + 0.201·15-s − 4.90·16-s − 5.73·17-s + 2.58·18-s + 4.22·19-s − 0.210·20-s − 5.42·21-s + 0.269·22-s − 3.75·23-s − 1.57·24-s − 4.97·25-s − 3.06·26-s + 5.55·27-s + 5.65·28-s + ⋯
L(s)  = 1  − 1.28·2-s − 0.725·3-s + 0.654·4-s − 0.0717·5-s + 0.932·6-s + 1.63·7-s + 0.443·8-s − 0.474·9-s + 0.0922·10-s − 0.0447·11-s − 0.474·12-s + 0.467·13-s − 2.09·14-s + 0.0520·15-s − 1.22·16-s − 1.39·17-s + 0.610·18-s + 0.970·19-s − 0.0469·20-s − 1.18·21-s + 0.0575·22-s − 0.782·23-s − 0.321·24-s − 0.994·25-s − 0.601·26-s + 1.06·27-s + 1.06·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-1$
Analytic conductor: \(14.7643\)
Root analytic conductor: \(3.84243\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1849,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + 1.81T + 2T^{2} \)
3 \( 1 + 1.25T + 3T^{2} \)
5 \( 1 + 0.160T + 5T^{2} \)
7 \( 1 - 4.31T + 7T^{2} \)
11 \( 1 + 0.148T + 11T^{2} \)
13 \( 1 - 1.68T + 13T^{2} \)
17 \( 1 + 5.73T + 17T^{2} \)
19 \( 1 - 4.22T + 19T^{2} \)
23 \( 1 + 3.75T + 23T^{2} \)
29 \( 1 + 8.83T + 29T^{2} \)
31 \( 1 - 8.64T + 31T^{2} \)
37 \( 1 + 7.19T + 37T^{2} \)
41 \( 1 + 3.67T + 41T^{2} \)
47 \( 1 + 2.13T + 47T^{2} \)
53 \( 1 - 1.98T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 5.65T + 61T^{2} \)
67 \( 1 + 1.42T + 67T^{2} \)
71 \( 1 - 7.56T + 71T^{2} \)
73 \( 1 - 4.99T + 73T^{2} \)
79 \( 1 + 14.8T + 79T^{2} \)
83 \( 1 + 6.11T + 83T^{2} \)
89 \( 1 - 8.33T + 89T^{2} \)
97 \( 1 - 2.10T + 97T^{2} \)
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

−8.660000833788523673408286295975, −8.266846752439302018292357317386, −7.55280417311951547081242580011, −6.66919183095750197863815175669, −5.60823757249561454995325135098, −4.90348580425042509243342455655, −3.99900206044438133759396830129, −2.23944654455986462378402611645, −1.34509169526705467464585047458, 0, 1.34509169526705467464585047458, 2.23944654455986462378402611645, 3.99900206044438133759396830129, 4.90348580425042509243342455655, 5.60823757249561454995325135098, 6.66919183095750197863815175669, 7.55280417311951547081242580011, 8.266846752439302018292357317386, 8.660000833788523673408286295975

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