Properties

Label 2-43e2-1.1-c1-0-115
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  + 2.31·2-s + 2.69·3-s + 3.34·4-s + 2.53·5-s + 6.24·6-s − 2.18·7-s + 3.11·8-s + 4.28·9-s + 5.87·10-s − 3.12·11-s + 9.03·12-s + 3.44·13-s − 5.05·14-s + 6.85·15-s + 0.513·16-s − 2.33·17-s + 9.91·18-s + 3.38·19-s + 8.50·20-s − 5.90·21-s − 7.21·22-s − 6.58·23-s + 8.41·24-s + 1.45·25-s + 7.96·26-s + 3.48·27-s − 7.32·28-s + ⋯
L(s)  = 1  + 1.63·2-s + 1.55·3-s + 1.67·4-s + 1.13·5-s + 2.54·6-s − 0.826·7-s + 1.10·8-s + 1.42·9-s + 1.85·10-s − 0.941·11-s + 2.60·12-s + 0.954·13-s − 1.35·14-s + 1.77·15-s + 0.128·16-s − 0.566·17-s + 2.33·18-s + 0.775·19-s + 1.90·20-s − 1.28·21-s − 1.53·22-s − 1.37·23-s + 1.71·24-s + 0.290·25-s + 1.56·26-s + 0.669·27-s − 1.38·28-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

\(L(1)\) \(\approx\) \(7.781402201\)
\(L(\frac12)\) \(\approx\) \(7.781402201\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.31T + 2T^{2} \)
3 \( 1 - 2.69T + 3T^{2} \)
5 \( 1 - 2.53T + 5T^{2} \)
7 \( 1 + 2.18T + 7T^{2} \)
11 \( 1 + 3.12T + 11T^{2} \)
13 \( 1 - 3.44T + 13T^{2} \)
17 \( 1 + 2.33T + 17T^{2} \)
19 \( 1 - 3.38T + 19T^{2} \)
23 \( 1 + 6.58T + 23T^{2} \)
29 \( 1 + 5.41T + 29T^{2} \)
31 \( 1 - 2.13T + 31T^{2} \)
37 \( 1 + 3.47T + 37T^{2} \)
41 \( 1 - 2.13T + 41T^{2} \)
47 \( 1 + 1.16T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 - 6.14T + 59T^{2} \)
61 \( 1 + 2.50T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 8.36T + 73T^{2} \)
79 \( 1 - 9.81T + 79T^{2} \)
83 \( 1 - 2.86T + 83T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + 4.22T + 97T^{2} \)
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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

−9.324212205136658145131655724324, −8.450425674045424727166901629738, −7.55337460763064742913706117154, −6.60199246766426354633260390298, −5.90887215682536454398684771582, −5.20498028237316628173727480025, −3.94889635264386704515080765800, −3.42935008150374285646485545374, −2.51871565341254824095429763170, −1.97159110648287562604990427649, 1.97159110648287562604990427649, 2.51871565341254824095429763170, 3.42935008150374285646485545374, 3.94889635264386704515080765800, 5.20498028237316628173727480025, 5.90887215682536454398684771582, 6.60199246766426354633260390298, 7.55337460763064742913706117154, 8.450425674045424727166901629738, 9.324212205136658145131655724324

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