Properties

Label 2-2898-1.1-c1-0-47
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 − 2.56·5-s − 7-s + 8-s − 2.56·10-s + 3.12·11-s + 0.561·13-s − 14-s + 16-s − 7.12·17-s + 3.12·19-s − 2.56·20-s + 3.12·22-s − 23-s + 1.56·25-s + 0.561·26-s − 28-s − 3.43·29-s − 5.12·31-s + 32-s − 7.12·34-s + 2.56·35-s + 2.56·37-s + 3.12·38-s − 2.56·40-s + 0.561·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.14·5-s − 0.377·7-s + 0.353·8-s − 0.810·10-s + 0.941·11-s + 0.155·13-s − 0.267·14-s + 0.250·16-s − 1.72·17-s + 0.716·19-s − 0.572·20-s + 0.665·22-s − 0.208·23-s + 0.312·25-s + 0.110·26-s − 0.188·28-s − 0.638·29-s − 0.920·31-s + 0.176·32-s − 1.22·34-s + 0.432·35-s + 0.421·37-s + 0.506·38-s − 0.405·40-s + 0.0876·41-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 \( 1 + 2.56T + 5T^{2} \)
11 \( 1 - 3.12T + 11T^{2} \)
13 \( 1 - 0.561T + 13T^{2} \)
17 \( 1 + 7.12T + 17T^{2} \)
19 \( 1 - 3.12T + 19T^{2} \)
29 \( 1 + 3.43T + 29T^{2} \)
31 \( 1 + 5.12T + 31T^{2} \)
37 \( 1 - 2.56T + 37T^{2} \)
41 \( 1 - 0.561T + 41T^{2} \)
43 \( 1 + 5.68T + 43T^{2} \)
47 \( 1 + 6.56T + 47T^{2} \)
53 \( 1 - 2.24T + 53T^{2} \)
59 \( 1 + 13.1T + 59T^{2} \)
61 \( 1 - 9.12T + 61T^{2} \)
67 \( 1 - 7.12T + 67T^{2} \)
71 \( 1 + 15.3T + 71T^{2} \)
73 \( 1 - 0.876T + 73T^{2} \)
79 \( 1 + 2.24T + 79T^{2} \)
83 \( 1 - 0.876T + 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + 16.5T + 97T^{2} \)
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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

−8.328872305181468231792344693402, −7.45061820328268051734395708104, −6.84437884226441358308352963666, −6.18020019111576913208434725983, −5.16837199052561964432297922465, −4.19864030745313188478856111238, −3.82376625783524314410067799514, −2.88593429190987833786608650985, −1.63328528299370640231478682293, 0, 1.63328528299370640231478682293, 2.88593429190987833786608650985, 3.82376625783524314410067799514, 4.19864030745313188478856111238, 5.16837199052561964432297922465, 6.18020019111576913208434725983, 6.84437884226441358308352963666, 7.45061820328268051734395708104, 8.328872305181468231792344693402

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