Properties

Label 2-43e2-1.1-c1-0-54
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.18·2-s + 2.30·3-s − 0.595·4-s + 3.24·5-s − 2.72·6-s − 2.76·7-s + 3.07·8-s + 2.30·9-s − 3.84·10-s + 5.58·11-s − 1.37·12-s + 1.64·13-s + 3.28·14-s + 7.46·15-s − 2.45·16-s − 0.758·17-s − 2.72·18-s + 2.17·19-s − 1.93·20-s − 6.37·21-s − 6.61·22-s − 0.00141·23-s + 7.08·24-s + 5.51·25-s − 1.94·26-s − 1.61·27-s + 1.64·28-s + ⋯
L(s)  = 1  − 0.838·2-s + 1.32·3-s − 0.297·4-s + 1.45·5-s − 1.11·6-s − 1.04·7-s + 1.08·8-s + 0.766·9-s − 1.21·10-s + 1.68·11-s − 0.395·12-s + 0.455·13-s + 0.876·14-s + 1.92·15-s − 0.613·16-s − 0.183·17-s − 0.642·18-s + 0.499·19-s − 0.431·20-s − 1.39·21-s − 1.41·22-s − 0.000294·23-s + 1.44·24-s + 1.10·25-s − 0.382·26-s − 0.309·27-s + 0.311·28-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: \(0\)
Selberg data: \((2,\ 1849,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.120588057\)
\(L(\frac12)\) \(\approx\) \(2.120588057\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.18T + 2T^{2} \)
3 \( 1 - 2.30T + 3T^{2} \)
5 \( 1 - 3.24T + 5T^{2} \)
7 \( 1 + 2.76T + 7T^{2} \)
11 \( 1 - 5.58T + 11T^{2} \)
13 \( 1 - 1.64T + 13T^{2} \)
17 \( 1 + 0.758T + 17T^{2} \)
19 \( 1 - 2.17T + 19T^{2} \)
23 \( 1 + 0.00141T + 23T^{2} \)
29 \( 1 - 4.68T + 29T^{2} \)
31 \( 1 - 4.03T + 31T^{2} \)
37 \( 1 + 1.99T + 37T^{2} \)
41 \( 1 + 7.00T + 41T^{2} \)
47 \( 1 + 7.86T + 47T^{2} \)
53 \( 1 + 3.94T + 53T^{2} \)
59 \( 1 - 3.64T + 59T^{2} \)
61 \( 1 - 8.55T + 61T^{2} \)
67 \( 1 - 9.38T + 67T^{2} \)
71 \( 1 - 6.90T + 71T^{2} \)
73 \( 1 - 11.1T + 73T^{2} \)
79 \( 1 - 6.53T + 79T^{2} \)
83 \( 1 + 5.70T + 83T^{2} \)
89 \( 1 - 8.63T + 89T^{2} \)
97 \( 1 + 17.6T + 97T^{2} \)
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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

−9.336573304865750289561814551503, −8.676320326962922533946875248488, −8.119066907395666639461108185842, −6.76787852319248455491743545385, −6.47385670812629217135287350877, −5.22672425418197926373951267202, −3.97470657287586115222477194670, −3.21357444296843078670409368836, −2.07098692501878898073120663100, −1.17241130452921591703815936557, 1.17241130452921591703815936557, 2.07098692501878898073120663100, 3.21357444296843078670409368836, 3.97470657287586115222477194670, 5.22672425418197926373951267202, 6.47385670812629217135287350877, 6.76787852319248455491743545385, 8.119066907395666639461108185842, 8.676320326962922533946875248488, 9.336573304865750289561814551503

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