Properties

Label 2-41e2-1.1-c1-0-23
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  + 0.305·2-s − 1.36·3-s − 1.90·4-s + 3.46·5-s − 0.417·6-s − 1.45·7-s − 1.19·8-s − 1.13·9-s + 1.06·10-s − 1.82·11-s + 2.60·12-s − 6.40·13-s − 0.445·14-s − 4.72·15-s + 3.44·16-s + 3.18·17-s − 0.348·18-s + 3.34·19-s − 6.60·20-s + 1.98·21-s − 0.558·22-s + 5.41·23-s + 1.63·24-s + 7.01·25-s − 1.96·26-s + 5.64·27-s + 2.77·28-s + ⋯
L(s)  = 1  + 0.216·2-s − 0.787·3-s − 0.953·4-s + 1.55·5-s − 0.170·6-s − 0.550·7-s − 0.422·8-s − 0.379·9-s + 0.335·10-s − 0.550·11-s + 0.750·12-s − 1.77·13-s − 0.119·14-s − 1.22·15-s + 0.861·16-s + 0.771·17-s − 0.0820·18-s + 0.767·19-s − 1.47·20-s + 0.433·21-s − 0.119·22-s + 1.12·23-s + 0.332·24-s + 1.40·25-s − 0.384·26-s + 1.08·27-s + 0.524·28-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

\(L(1)\) \(\approx\) \(1.085577252\)
\(L(\frac12)\) \(\approx\) \(1.085577252\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.305T + 2T^{2} \)
3 \( 1 + 1.36T + 3T^{2} \)
5 \( 1 - 3.46T + 5T^{2} \)
7 \( 1 + 1.45T + 7T^{2} \)
11 \( 1 + 1.82T + 11T^{2} \)
13 \( 1 + 6.40T + 13T^{2} \)
17 \( 1 - 3.18T + 17T^{2} \)
19 \( 1 - 3.34T + 19T^{2} \)
23 \( 1 - 5.41T + 23T^{2} \)
29 \( 1 + 0.339T + 29T^{2} \)
31 \( 1 + 3.75T + 31T^{2} \)
37 \( 1 + 1.22T + 37T^{2} \)
43 \( 1 - 7.39T + 43T^{2} \)
47 \( 1 - 1.58T + 47T^{2} \)
53 \( 1 + 1.14T + 53T^{2} \)
59 \( 1 - 13.4T + 59T^{2} \)
61 \( 1 - 4.62T + 61T^{2} \)
67 \( 1 - 6.67T + 67T^{2} \)
71 \( 1 + 8.26T + 71T^{2} \)
73 \( 1 - 8.99T + 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + 7.11T + 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 + 3.28T + 97T^{2} \)
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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

−9.547742866806149315613026720386, −8.873849197353508666423993229509, −7.67067225053065712518072422935, −6.74933905763283823401395649377, −5.75574453176568988939922028874, −5.32078962164261456362524058709, −4.86297041843553586213497445594, −3.27748897891394692335357767940, −2.40198488623230958916521455176, −0.71501326309398181769609833026, 0.71501326309398181769609833026, 2.40198488623230958916521455176, 3.27748897891394692335357767940, 4.86297041843553586213497445594, 5.32078962164261456362524058709, 5.75574453176568988939922028874, 6.74933905763283823401395649377, 7.67067225053065712518072422935, 8.873849197353508666423993229509, 9.547742866806149315613026720386

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