Properties

Label 2-153-1.1-c3-0-13
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  + 5.03·2-s + 17.3·4-s − 0.885·5-s + 3.81·7-s + 46.9·8-s − 4.45·10-s + 52.3·11-s − 8.06·13-s + 19.2·14-s + 97.5·16-s + 17·17-s − 66.5·19-s − 15.3·20-s + 263.·22-s − 180.·23-s − 124.·25-s − 40.5·26-s + 66.1·28-s + 41.2·29-s − 34.9·31-s + 115.·32-s + 85.5·34-s − 3.38·35-s + 130.·37-s − 334.·38-s − 41.5·40-s + 17.9·41-s + ⋯
L(s)  = 1  + 1.77·2-s + 2.16·4-s − 0.0792·5-s + 0.206·7-s + 2.07·8-s − 0.140·10-s + 1.43·11-s − 0.171·13-s + 0.366·14-s + 1.52·16-s + 0.242·17-s − 0.803·19-s − 0.171·20-s + 2.55·22-s − 1.63·23-s − 0.993·25-s − 0.305·26-s + 0.446·28-s + 0.264·29-s − 0.202·31-s + 0.638·32-s + 0.431·34-s − 0.0163·35-s + 0.579·37-s − 1.42·38-s − 0.164·40-s + 0.0682·41-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

\(L(2)\) \(\approx\) \(4.660620220\)
\(L(\frac12)\) \(\approx\) \(4.660620220\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.03T + 8T^{2} \)
5 \( 1 + 0.885T + 125T^{2} \)
7 \( 1 - 3.81T + 343T^{2} \)
11 \( 1 - 52.3T + 1.33e3T^{2} \)
13 \( 1 + 8.06T + 2.19e3T^{2} \)
19 \( 1 + 66.5T + 6.85e3T^{2} \)
23 \( 1 + 180.T + 1.21e4T^{2} \)
29 \( 1 - 41.2T + 2.43e4T^{2} \)
31 \( 1 + 34.9T + 2.97e4T^{2} \)
37 \( 1 - 130.T + 5.06e4T^{2} \)
41 \( 1 - 17.9T + 6.89e4T^{2} \)
43 \( 1 - 277.T + 7.95e4T^{2} \)
47 \( 1 + 463.T + 1.03e5T^{2} \)
53 \( 1 - 329.T + 1.48e5T^{2} \)
59 \( 1 + 678.T + 2.05e5T^{2} \)
61 \( 1 - 340.T + 2.26e5T^{2} \)
67 \( 1 - 15.3T + 3.00e5T^{2} \)
71 \( 1 - 670.T + 3.57e5T^{2} \)
73 \( 1 - 193.T + 3.89e5T^{2} \)
79 \( 1 - 1.08e3T + 4.93e5T^{2} \)
83 \( 1 - 865.T + 5.71e5T^{2} \)
89 \( 1 + 1.12e3T + 7.04e5T^{2} \)
97 \( 1 + 379.T + 9.12e5T^{2} \)
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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

−12.48489624373236775736123338260, −11.88052770256101306632685248067, −11.02737283016101839651636562940, −9.624396572606530069461395954809, −8.002435324201369610023989484499, −6.65283483138256497126441004186, −5.87803139257014233075472377117, −4.47662001400708767625410092592, −3.66187611369757625938922168409, −1.98678942895908199373291215876, 1.98678942895908199373291215876, 3.66187611369757625938922168409, 4.47662001400708767625410092592, 5.87803139257014233075472377117, 6.65283483138256497126441004186, 8.002435324201369610023989484499, 9.624396572606530069461395954809, 11.02737283016101839651636562940, 11.88052770256101306632685248067, 12.48489624373236775736123338260

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