Properties

Label 2-153-1.1-c3-0-18
Degree $2$
Conductor $153$
Sign $-1$
Analytic cond. $9.02729$
Root an. cond. $3.00454$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 4-s − 6·5-s − 28·7-s − 21·8-s − 18·10-s + 24·11-s − 58·13-s − 84·14-s − 71·16-s − 17·17-s + 116·19-s − 6·20-s + 72·22-s + 60·23-s − 89·25-s − 174·26-s − 28·28-s − 30·29-s − 172·31-s − 45·32-s − 51·34-s + 168·35-s − 58·37-s + 348·38-s + 126·40-s + 342·41-s + ⋯
L(s)  = 1  + 1.06·2-s + 1/8·4-s − 0.536·5-s − 1.51·7-s − 0.928·8-s − 0.569·10-s + 0.657·11-s − 1.23·13-s − 1.60·14-s − 1.10·16-s − 0.242·17-s + 1.40·19-s − 0.0670·20-s + 0.697·22-s + 0.543·23-s − 0.711·25-s − 1.31·26-s − 0.188·28-s − 0.192·29-s − 0.996·31-s − 0.248·32-s − 0.257·34-s + 0.811·35-s − 0.257·37-s + 1.48·38-s + 0.498·40-s + 1.30·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(9.02729\)
Root analytic conductor: \(3.00454\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 153,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
good2 \( 1 - 3 T + p^{3} T^{2} \)
5 \( 1 + 6 T + p^{3} T^{2} \)
7 \( 1 + 4 p T + p^{3} T^{2} \)
11 \( 1 - 24 T + p^{3} T^{2} \)
13 \( 1 + 58 T + p^{3} T^{2} \)
19 \( 1 - 116 T + p^{3} T^{2} \)
23 \( 1 - 60 T + p^{3} T^{2} \)
29 \( 1 + 30 T + p^{3} T^{2} \)
31 \( 1 + 172 T + p^{3} T^{2} \)
37 \( 1 + 58 T + p^{3} T^{2} \)
41 \( 1 - 342 T + p^{3} T^{2} \)
43 \( 1 + 148 T + p^{3} T^{2} \)
47 \( 1 + 288 T + p^{3} T^{2} \)
53 \( 1 + 6 p T + p^{3} T^{2} \)
59 \( 1 + 252 T + p^{3} T^{2} \)
61 \( 1 - 110 T + p^{3} T^{2} \)
67 \( 1 + 484 T + p^{3} T^{2} \)
71 \( 1 - 708 T + p^{3} T^{2} \)
73 \( 1 - 362 T + p^{3} T^{2} \)
79 \( 1 + 484 T + p^{3} T^{2} \)
83 \( 1 + 756 T + p^{3} T^{2} \)
89 \( 1 - 774 T + p^{3} T^{2} \)
97 \( 1 + 382 T + p^{3} T^{2} \)
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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

−12.29832645046066653290861761855, −11.44486693547653239374504329741, −9.747235020419909570129643868121, −9.201717554277108621744918319431, −7.44341663706080716828933604201, −6.41269876808547644856589628353, −5.21651946848089499214866262266, −3.89996217873455686527651949802, −2.98219859425725005523103074571, 0, 2.98219859425725005523103074571, 3.89996217873455686527651949802, 5.21651946848089499214866262266, 6.41269876808547644856589628353, 7.44341663706080716828933604201, 9.201717554277108621744918319431, 9.747235020419909570129643868121, 11.44486693547653239374504329741, 12.29832645046066653290861761855

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