Properties

Label 2-43e2-1.1-c1-0-121
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.42·2-s − 1.86·3-s + 3.87·4-s + 2.06·5-s − 4.51·6-s − 4.60·7-s + 4.55·8-s + 0.464·9-s + 5.00·10-s − 2.98·11-s − 7.22·12-s + 0.123·13-s − 11.1·14-s − 3.83·15-s + 3.29·16-s − 1.25·17-s + 1.12·18-s − 1.44·19-s + 8.00·20-s + 8.57·21-s − 7.23·22-s − 5.31·23-s − 8.48·24-s − 0.747·25-s + 0.298·26-s + 4.71·27-s − 17.8·28-s + ⋯
L(s)  = 1  + 1.71·2-s − 1.07·3-s + 1.93·4-s + 0.922·5-s − 1.84·6-s − 1.74·7-s + 1.61·8-s + 0.154·9-s + 1.58·10-s − 0.900·11-s − 2.08·12-s + 0.0341·13-s − 2.98·14-s − 0.991·15-s + 0.823·16-s − 0.304·17-s + 0.265·18-s − 0.330·19-s + 1.78·20-s + 1.87·21-s − 1.54·22-s − 1.10·23-s − 1.73·24-s − 0.149·25-s + 0.0585·26-s + 0.908·27-s − 3.37·28-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: \(1\)
Selberg data: \((2,\ 1849,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.42T + 2T^{2} \)
3 \( 1 + 1.86T + 3T^{2} \)
5 \( 1 - 2.06T + 5T^{2} \)
7 \( 1 + 4.60T + 7T^{2} \)
11 \( 1 + 2.98T + 11T^{2} \)
13 \( 1 - 0.123T + 13T^{2} \)
17 \( 1 + 1.25T + 17T^{2} \)
19 \( 1 + 1.44T + 19T^{2} \)
23 \( 1 + 5.31T + 23T^{2} \)
29 \( 1 - 0.662T + 29T^{2} \)
31 \( 1 + 9.01T + 31T^{2} \)
37 \( 1 - 6.93T + 37T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
47 \( 1 + 0.503T + 47T^{2} \)
53 \( 1 + 2.70T + 53T^{2} \)
59 \( 1 + 8.70T + 59T^{2} \)
61 \( 1 + 2.85T + 61T^{2} \)
67 \( 1 + 8.40T + 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + 0.0416T + 73T^{2} \)
79 \( 1 - 4.41T + 79T^{2} \)
83 \( 1 - 4.42T + 83T^{2} \)
89 \( 1 + 12.9T + 89T^{2} \)
97 \( 1 + 2.27T + 97T^{2} \)
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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

−9.115852438356746721375837209083, −7.59176285346382730380052830103, −6.61979335926795269132161708190, −6.02204669947839427248275401771, −5.83007882410201572060681078909, −4.95741694429012570773866858671, −3.94892943321281431203180891166, −2.98963015485249900481974771114, −2.17819099609554718236809097250, 0, 2.17819099609554718236809097250, 2.98963015485249900481974771114, 3.94892943321281431203180891166, 4.95741694429012570773866858671, 5.83007882410201572060681078909, 6.02204669947839427248275401771, 6.61979335926795269132161708190, 7.59176285346382730380052830103, 9.115852438356746721375837209083

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