Properties

Label 2-153-1.1-c3-0-5
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4.67·2-s + 13.8·4-s + 11.9·5-s + 26.1·7-s − 27.1·8-s − 55.6·10-s + 3.24·11-s − 20.0·13-s − 122.·14-s + 16.4·16-s + 17·17-s + 57.3·19-s + 164.·20-s − 15.1·22-s − 77.0·23-s + 17.0·25-s + 93.6·26-s + 361.·28-s + 286.·29-s − 8.54·31-s + 140.·32-s − 79.4·34-s + 311.·35-s + 357.·37-s − 267.·38-s − 324.·40-s − 194.·41-s + ⋯
L(s)  = 1  − 1.65·2-s + 1.72·4-s + 1.06·5-s + 1.41·7-s − 1.20·8-s − 1.76·10-s + 0.0889·11-s − 0.427·13-s − 2.32·14-s + 0.257·16-s + 0.242·17-s + 0.692·19-s + 1.84·20-s − 0.146·22-s − 0.698·23-s + 0.136·25-s + 0.706·26-s + 2.43·28-s + 1.83·29-s − 0.0495·31-s + 0.777·32-s − 0.400·34-s + 1.50·35-s + 1.59·37-s − 1.14·38-s − 1.28·40-s − 0.740·41-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

\(L(2)\) \(\approx\) \(1.077215835\)
\(L(\frac12)\) \(\approx\) \(1.077215835\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.67T + 8T^{2} \)
5 \( 1 - 11.9T + 125T^{2} \)
7 \( 1 - 26.1T + 343T^{2} \)
11 \( 1 - 3.24T + 1.33e3T^{2} \)
13 \( 1 + 20.0T + 2.19e3T^{2} \)
19 \( 1 - 57.3T + 6.85e3T^{2} \)
23 \( 1 + 77.0T + 1.21e4T^{2} \)
29 \( 1 - 286.T + 2.43e4T^{2} \)
31 \( 1 + 8.54T + 2.97e4T^{2} \)
37 \( 1 - 357.T + 5.06e4T^{2} \)
41 \( 1 + 194.T + 6.89e4T^{2} \)
43 \( 1 + 74.2T + 7.95e4T^{2} \)
47 \( 1 + 23.6T + 1.03e5T^{2} \)
53 \( 1 + 104.T + 1.48e5T^{2} \)
59 \( 1 + 249.T + 2.05e5T^{2} \)
61 \( 1 + 370.T + 2.26e5T^{2} \)
67 \( 1 - 939.T + 3.00e5T^{2} \)
71 \( 1 - 520.T + 3.57e5T^{2} \)
73 \( 1 - 348.T + 3.89e5T^{2} \)
79 \( 1 + 953.T + 4.93e5T^{2} \)
83 \( 1 - 1.41e3T + 5.71e5T^{2} \)
89 \( 1 - 486.T + 7.04e5T^{2} \)
97 \( 1 + 685.T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.06355655455183979112984206493, −11.17733506649139507468506665512, −10.17207865755284213327310921791, −9.526513657271677198750921965248, −8.384103942534636298713570385177, −7.66339150973919625829796199995, −6.33393508384993200101546082744, −4.92366682842881873154332931004, −2.30137169809090431660128705933, −1.19248705018502435910495260381, 1.19248705018502435910495260381, 2.30137169809090431660128705933, 4.92366682842881873154332931004, 6.33393508384993200101546082744, 7.66339150973919625829796199995, 8.384103942534636298713570385177, 9.526513657271677198750921965248, 10.17207865755284213327310921791, 11.17733506649139507468506665512, 12.06355655455183979112984206493

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