Properties

Label 2-153-1.1-c1-0-2
Degree $2$
Conductor $153$
Sign $-1$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 5-s − 2·7-s + 2·10-s − 3·11-s − 5·13-s + 4·14-s − 4·16-s − 17-s − 19-s − 2·20-s + 6·22-s − 7·23-s − 4·25-s + 10·26-s − 4·28-s + 6·29-s + 4·31-s + 8·32-s + 2·34-s + 2·35-s + 10·37-s + 2·38-s + 9·41-s + 43-s − 6·44-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.447·5-s − 0.755·7-s + 0.632·10-s − 0.904·11-s − 1.38·13-s + 1.06·14-s − 16-s − 0.242·17-s − 0.229·19-s − 0.447·20-s + 1.27·22-s − 1.45·23-s − 4/5·25-s + 1.96·26-s − 0.755·28-s + 1.11·29-s + 0.718·31-s + 1.41·32-s + 0.342·34-s + 0.338·35-s + 1.64·37-s + 0.324·38-s + 1.40·41-s + 0.152·43-s − 0.904·44-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 153,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
17 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−12.29040567291484407710402327749, −11.21201107323355823458653573304, −9.992759799763264145998704328660, −9.673767858002872067860717754239, −8.193497167062057609442371334624, −7.62717436015796243936248846504, −6.35321842859344777216335456735, −4.54009421268158840480926538889, −2.52748238071021851087833645012, 0, 2.52748238071021851087833645012, 4.54009421268158840480926538889, 6.35321842859344777216335456735, 7.62717436015796243936248846504, 8.193497167062057609442371334624, 9.673767858002872067860717754239, 9.992759799763264145998704328660, 11.21201107323355823458653573304, 12.29040567291484407710402327749

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