Properties

Label 2-1881-1.1-c1-0-62
Degree $2$
Conductor $1881$
Sign $1$
Analytic cond. $15.0198$
Root an. cond. $3.87554$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.18·2-s + 2.77·4-s + 3.43·5-s + 3.93·7-s + 1.69·8-s + 7.49·10-s − 11-s + 3.31·13-s + 8.60·14-s − 1.84·16-s − 2.80·17-s − 19-s + 9.52·20-s − 2.18·22-s − 6.88·23-s + 6.77·25-s + 7.23·26-s + 10.9·28-s − 5.67·29-s + 2.51·31-s − 7.42·32-s − 6.13·34-s + 13.5·35-s − 6.39·37-s − 2.18·38-s + 5.81·40-s − 0.560·41-s + ⋯
L(s)  = 1  + 1.54·2-s + 1.38·4-s + 1.53·5-s + 1.48·7-s + 0.598·8-s + 2.37·10-s − 0.301·11-s + 0.918·13-s + 2.30·14-s − 0.462·16-s − 0.680·17-s − 0.229·19-s + 2.12·20-s − 0.465·22-s − 1.43·23-s + 1.35·25-s + 1.41·26-s + 2.06·28-s − 1.05·29-s + 0.452·31-s − 1.31·32-s − 1.05·34-s + 2.28·35-s − 1.05·37-s − 0.354·38-s + 0.919·40-s − 0.0875·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1881\)    =    \(3^{2} \cdot 11 \cdot 19\)
Sign: $1$
Analytic conductor: \(15.0198\)
Root analytic conductor: \(3.87554\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1881,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.901925976\)
\(L(\frac12)\) \(\approx\) \(5.901925976\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + T \)
19 \( 1 + T \)
good2 \( 1 - 2.18T + 2T^{2} \)
5 \( 1 - 3.43T + 5T^{2} \)
7 \( 1 - 3.93T + 7T^{2} \)
13 \( 1 - 3.31T + 13T^{2} \)
17 \( 1 + 2.80T + 17T^{2} \)
23 \( 1 + 6.88T + 23T^{2} \)
29 \( 1 + 5.67T + 29T^{2} \)
31 \( 1 - 2.51T + 31T^{2} \)
37 \( 1 + 6.39T + 37T^{2} \)
41 \( 1 + 0.560T + 41T^{2} \)
43 \( 1 + 9.40T + 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 + 5.68T + 53T^{2} \)
59 \( 1 + 4.35T + 59T^{2} \)
61 \( 1 + 3.56T + 61T^{2} \)
67 \( 1 + 9.95T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 8.95T + 73T^{2} \)
79 \( 1 - 8.49T + 79T^{2} \)
83 \( 1 - 5.21T + 83T^{2} \)
89 \( 1 - 7.28T + 89T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.147682039713645139706708897810, −8.474167875518717747717093912203, −7.44315232297323850331442867219, −6.31042616935053828486743537464, −5.90754215271266643690424027762, −5.12696848381543459067539136106, −4.52085708578687058454735564483, −3.50562973823269645177866913958, −2.17569141872391864010893096727, −1.76359882748356636509338428509, 1.76359882748356636509338428509, 2.17569141872391864010893096727, 3.50562973823269645177866913958, 4.52085708578687058454735564483, 5.12696848381543459067539136106, 5.90754215271266643690424027762, 6.31042616935053828486743537464, 7.44315232297323850331442867219, 8.474167875518717747717093912203, 9.147682039713645139706708897810

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