Properties

Label 2-153-1.1-c3-0-8
Degree $2$
Conductor $153$
Sign $1$
Analytic cond. $9.02729$
Root an. cond. $3.00454$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.47·2-s + 4.08·4-s + 7.60·5-s + 16.3·7-s − 13.6·8-s + 26.4·10-s + 42.4·11-s + 62.2·13-s + 56.9·14-s − 79.9·16-s − 17·17-s − 52.0·19-s + 31.0·20-s + 147.·22-s + 67.8·23-s − 67.1·25-s + 216.·26-s + 67.0·28-s + 91.7·29-s − 177.·31-s − 169.·32-s − 59.1·34-s + 124.·35-s − 258.·37-s − 181.·38-s − 103.·40-s + 262.·41-s + ⋯
L(s)  = 1  + 1.22·2-s + 0.511·4-s + 0.680·5-s + 0.885·7-s − 0.601·8-s + 0.836·10-s + 1.16·11-s + 1.32·13-s + 1.08·14-s − 1.24·16-s − 0.242·17-s − 0.629·19-s + 0.347·20-s + 1.42·22-s + 0.615·23-s − 0.537·25-s + 1.63·26-s + 0.452·28-s + 0.587·29-s − 1.02·31-s − 0.935·32-s − 0.298·34-s + 0.602·35-s − 1.14·37-s − 0.773·38-s − 0.408·40-s + 0.998·41-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.553574526\)
\(L(\frac12)\) \(\approx\) \(3.553574526\)
\(L(\frac{5}{2})\) 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 \( 1 + 17T \)
good2 \( 1 - 3.47T + 8T^{2} \)
5 \( 1 - 7.60T + 125T^{2} \)
7 \( 1 - 16.3T + 343T^{2} \)
11 \( 1 - 42.4T + 1.33e3T^{2} \)
13 \( 1 - 62.2T + 2.19e3T^{2} \)
19 \( 1 + 52.0T + 6.85e3T^{2} \)
23 \( 1 - 67.8T + 1.21e4T^{2} \)
29 \( 1 - 91.7T + 2.43e4T^{2} \)
31 \( 1 + 177.T + 2.97e4T^{2} \)
37 \( 1 + 258.T + 5.06e4T^{2} \)
41 \( 1 - 262.T + 6.89e4T^{2} \)
43 \( 1 + 331.T + 7.95e4T^{2} \)
47 \( 1 + 49.0T + 1.03e5T^{2} \)
53 \( 1 + 462.T + 1.48e5T^{2} \)
59 \( 1 + 351.T + 2.05e5T^{2} \)
61 \( 1 - 41.6T + 2.26e5T^{2} \)
67 \( 1 - 845.T + 3.00e5T^{2} \)
71 \( 1 + 420.T + 3.57e5T^{2} \)
73 \( 1 + 402.T + 3.89e5T^{2} \)
79 \( 1 + 736.T + 4.93e5T^{2} \)
83 \( 1 - 1.24e3T + 5.71e5T^{2} \)
89 \( 1 - 463.T + 7.04e5T^{2} \)
97 \( 1 + 1.23e3T + 9.12e5T^{2} \)
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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

−12.74011078774311245884991315754, −11.65159954884225246772236373896, −10.88481840607633758307109417790, −9.323358673628956548676796627596, −8.467239196079302647020900765164, −6.67228043362126840036621343648, −5.82407569020152945177409053080, −4.65342662234711925541295199523, −3.55213125816418111606059672348, −1.70701678348170145307387506065, 1.70701678348170145307387506065, 3.55213125816418111606059672348, 4.65342662234711925541295199523, 5.82407569020152945177409053080, 6.67228043362126840036621343648, 8.467239196079302647020900765164, 9.323358673628956548676796627596, 10.88481840607633758307109417790, 11.65159954884225246772236373896, 12.74011078774311245884991315754

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