Properties

Label 2-41e2-1.1-c1-0-39
Degree $2$
Conductor $1681$
Sign $1$
Analytic cond. $13.4228$
Root an. cond. $3.66372$
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  − 1.71·2-s + 3.15·3-s + 0.938·4-s − 2.70·5-s − 5.40·6-s + 1.61·7-s + 1.81·8-s + 6.93·9-s + 4.64·10-s + 0.692·11-s + 2.95·12-s + 1.52·13-s − 2.76·14-s − 8.53·15-s − 4.99·16-s + 1.05·17-s − 11.8·18-s + 0.128·19-s − 2.54·20-s + 5.08·21-s − 1.18·22-s + 1.88·23-s + 5.73·24-s + 2.34·25-s − 2.61·26-s + 12.3·27-s + 1.51·28-s + ⋯
L(s)  = 1  − 1.21·2-s + 1.81·3-s + 0.469·4-s − 1.21·5-s − 2.20·6-s + 0.609·7-s + 0.643·8-s + 2.31·9-s + 1.46·10-s + 0.208·11-s + 0.854·12-s + 0.423·13-s − 0.738·14-s − 2.20·15-s − 1.24·16-s + 0.256·17-s − 2.80·18-s + 0.0294·19-s − 0.568·20-s + 1.10·21-s − 0.253·22-s + 0.392·23-s + 1.17·24-s + 0.468·25-s − 0.513·26-s + 2.38·27-s + 0.286·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1681\)    =    \(41^{2}\)
Sign: $1$
Analytic conductor: \(13.4228\)
Root analytic conductor: \(3.66372\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1681,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.590744907\)
\(L(\frac12)\) \(\approx\) \(1.590744907\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 \)
good2 \( 1 + 1.71T + 2T^{2} \)
3 \( 1 - 3.15T + 3T^{2} \)
5 \( 1 + 2.70T + 5T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 - 0.692T + 11T^{2} \)
13 \( 1 - 1.52T + 13T^{2} \)
17 \( 1 - 1.05T + 17T^{2} \)
19 \( 1 - 0.128T + 19T^{2} \)
23 \( 1 - 1.88T + 23T^{2} \)
29 \( 1 - 8.29T + 29T^{2} \)
31 \( 1 + 1.20T + 31T^{2} \)
37 \( 1 + 6.04T + 37T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 - 6.34T + 47T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 - 8.17T + 59T^{2} \)
61 \( 1 - 2.63T + 61T^{2} \)
67 \( 1 - 8.49T + 67T^{2} \)
71 \( 1 + 4.29T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 - 6.49T + 83T^{2} \)
89 \( 1 + 2.88T + 89T^{2} \)
97 \( 1 - 1.20T + 97T^{2} \)
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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

−8.973168164071908477844587259229, −8.463450804264049008836011546390, −8.179229697067864261790785750173, −7.40019809579394625365892117437, −6.86386686395836697584981493611, −4.88380765499791948991774980387, −4.05042283675877981108185062968, −3.32256886059397489460219167165, −2.11455625839298733684141352946, −1.03327292513117171975883558716, 1.03327292513117171975883558716, 2.11455625839298733684141352946, 3.32256886059397489460219167165, 4.05042283675877981108185062968, 4.88380765499791948991774980387, 6.86386686395836697584981493611, 7.40019809579394625365892117437, 8.179229697067864261790785750173, 8.463450804264049008836011546390, 8.973168164071908477844587259229

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