Properties

Label 2-153-1.1-c1-0-4
Degree $2$
Conductor $153$
Sign $1$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
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  + 2.56·2-s + 4.56·4-s − 3.56·5-s + 6.56·8-s − 9.12·10-s − 1.56·11-s + 0.438·13-s + 7.68·16-s − 17-s − 4.68·19-s − 16.2·20-s − 4·22-s + 2.43·23-s + 7.68·25-s + 1.12·26-s + 8.24·29-s + 3.12·31-s + 6.56·32-s − 2.56·34-s − 5.12·37-s − 11.9·38-s − 23.3·40-s + 3.56·41-s + 4.68·43-s − 7.12·44-s + 6.24·46-s + 11.1·47-s + ⋯
L(s)  = 1  + 1.81·2-s + 2.28·4-s − 1.59·5-s + 2.31·8-s − 2.88·10-s − 0.470·11-s + 0.121·13-s + 1.92·16-s − 0.242·17-s − 1.07·19-s − 3.63·20-s − 0.852·22-s + 0.508·23-s + 1.53·25-s + 0.220·26-s + 1.53·29-s + 0.560·31-s + 1.15·32-s − 0.439·34-s − 0.842·37-s − 1.94·38-s − 3.69·40-s + 0.556·41-s + 0.714·43-s − 1.07·44-s + 0.920·46-s + 1.62·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.333534695\)
\(L(\frac12)\) \(\approx\) \(2.333534695\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 + T \)
good2 \( 1 - 2.56T + 2T^{2} \)
5 \( 1 + 3.56T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 1.56T + 11T^{2} \)
13 \( 1 - 0.438T + 13T^{2} \)
19 \( 1 + 4.68T + 19T^{2} \)
23 \( 1 - 2.43T + 23T^{2} \)
29 \( 1 - 8.24T + 29T^{2} \)
31 \( 1 - 3.12T + 31T^{2} \)
37 \( 1 + 5.12T + 37T^{2} \)
41 \( 1 - 3.56T + 41T^{2} \)
43 \( 1 - 4.68T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + 7.12T + 59T^{2} \)
61 \( 1 - 9.12T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 6.24T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + 9.36T + 79T^{2} \)
83 \( 1 - 0.876T + 83T^{2} \)
89 \( 1 - 1.12T + 89T^{2} \)
97 \( 1 + 2.87T + 97T^{2} \)
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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

−12.81153383843558361801241281824, −12.26798724154686299614565601806, −11.33738186083091183668737219454, −10.62539255162187764834405087385, −8.452248548809288367614664414456, −7.37400369714000213035983918020, −6.35018764485142958760593183486, −4.85073210575943155801207233947, −4.06209328209427631305994086121, −2.86226885331132817253176394642, 2.86226885331132817253176394642, 4.06209328209427631305994086121, 4.85073210575943155801207233947, 6.35018764485142958760593183486, 7.37400369714000213035983918020, 8.452248548809288367614664414456, 10.62539255162187764834405087385, 11.33738186083091183668737219454, 12.26798724154686299614565601806, 12.81153383843558361801241281824

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