Properties

Label 2-2898-1.1-c1-0-55
Degree $2$
Conductor $2898$
Sign $-1$
Analytic cond. $23.1406$
Root an. cond. $4.81047$
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-s + 4-s + 1.56·5-s − 7-s + 8-s + 1.56·10-s − 5.12·11-s − 3.56·13-s − 14-s + 16-s + 1.12·17-s − 5.12·19-s + 1.56·20-s − 5.12·22-s − 23-s − 2.56·25-s − 3.56·26-s − 28-s − 7.56·29-s + 3.12·31-s + 32-s + 1.12·34-s − 1.56·35-s − 1.56·37-s − 5.12·38-s + 1.56·40-s − 3.56·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.698·5-s − 0.377·7-s + 0.353·8-s + 0.493·10-s − 1.54·11-s − 0.987·13-s − 0.267·14-s + 0.250·16-s + 0.272·17-s − 1.17·19-s + 0.349·20-s − 1.09·22-s − 0.208·23-s − 0.512·25-s − 0.698·26-s − 0.188·28-s − 1.40·29-s + 0.560·31-s + 0.176·32-s + 0.192·34-s − 0.263·35-s − 0.256·37-s − 0.831·38-s + 0.246·40-s − 0.556·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2898\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(23.1406\)
Root analytic conductor: \(4.81047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2898} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2898,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + T \)
23 \( 1 + T \)
good5 \( 1 - 1.56T + 5T^{2} \)
11 \( 1 + 5.12T + 11T^{2} \)
13 \( 1 + 3.56T + 13T^{2} \)
17 \( 1 - 1.12T + 17T^{2} \)
19 \( 1 + 5.12T + 19T^{2} \)
29 \( 1 + 7.56T + 29T^{2} \)
31 \( 1 - 3.12T + 31T^{2} \)
37 \( 1 + 1.56T + 37T^{2} \)
41 \( 1 + 3.56T + 41T^{2} \)
43 \( 1 - 6.68T + 43T^{2} \)
47 \( 1 + 2.43T + 47T^{2} \)
53 \( 1 + 14.2T + 53T^{2} \)
59 \( 1 + 4.87T + 59T^{2} \)
61 \( 1 - 0.876T + 61T^{2} \)
67 \( 1 + 1.12T + 67T^{2} \)
71 \( 1 - 9.36T + 71T^{2} \)
73 \( 1 - 9.12T + 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 - 9.12T + 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−8.137428607884339326372240436451, −7.64212240913575077579625388714, −6.70499397192025405259067539151, −5.96117013321385477602204595340, −5.29672860862609689290622670443, −4.62823137949894990766042509918, −3.53861950401774083328688677691, −2.55422667344097801323175761226, −1.96806931661533329899196232018, 0, 1.96806931661533329899196232018, 2.55422667344097801323175761226, 3.53861950401774083328688677691, 4.62823137949894990766042509918, 5.29672860862609689290622670443, 5.96117013321385477602204595340, 6.70499397192025405259067539151, 7.64212240913575077579625388714, 8.137428607884339326372240436451

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