Properties

Label 2-153-1.1-c3-0-2
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  − 1.36·2-s − 6.14·4-s − 3.03·5-s − 7.94·7-s + 19.2·8-s + 4.12·10-s − 27.6·11-s + 58.1·13-s + 10.8·14-s + 22.9·16-s + 17·17-s + 89.1·19-s + 18.6·20-s + 37.5·22-s + 115.·23-s − 115.·25-s − 79.1·26-s + 48.8·28-s + 128.·29-s + 273.·31-s − 185.·32-s − 23.1·34-s + 24.0·35-s − 132.·37-s − 121.·38-s − 58.3·40-s + 470.·41-s + ⋯
L(s)  = 1  − 0.481·2-s − 0.768·4-s − 0.271·5-s − 0.428·7-s + 0.851·8-s + 0.130·10-s − 0.756·11-s + 1.23·13-s + 0.206·14-s + 0.358·16-s + 0.242·17-s + 1.07·19-s + 0.208·20-s + 0.364·22-s + 1.04·23-s − 0.926·25-s − 0.596·26-s + 0.329·28-s + 0.823·29-s + 1.58·31-s − 1.02·32-s − 0.116·34-s + 0.116·35-s − 0.588·37-s − 0.518·38-s − 0.230·40-s + 1.79·41-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

\(L(2)\) \(\approx\) \(0.9558484037\)
\(L(\frac12)\) \(\approx\) \(0.9558484037\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.36T + 8T^{2} \)
5 \( 1 + 3.03T + 125T^{2} \)
7 \( 1 + 7.94T + 343T^{2} \)
11 \( 1 + 27.6T + 1.33e3T^{2} \)
13 \( 1 - 58.1T + 2.19e3T^{2} \)
19 \( 1 - 89.1T + 6.85e3T^{2} \)
23 \( 1 - 115.T + 1.21e4T^{2} \)
29 \( 1 - 128.T + 2.43e4T^{2} \)
31 \( 1 - 273.T + 2.97e4T^{2} \)
37 \( 1 + 132.T + 5.06e4T^{2} \)
41 \( 1 - 470.T + 6.89e4T^{2} \)
43 \( 1 - 352.T + 7.95e4T^{2} \)
47 \( 1 + 152.T + 1.03e5T^{2} \)
53 \( 1 + 527.T + 1.48e5T^{2} \)
59 \( 1 - 292.T + 2.05e5T^{2} \)
61 \( 1 + 53.8T + 2.26e5T^{2} \)
67 \( 1 - 52.9T + 3.00e5T^{2} \)
71 \( 1 + 788.T + 3.57e5T^{2} \)
73 \( 1 - 295.T + 3.89e5T^{2} \)
79 \( 1 + 720.T + 4.93e5T^{2} \)
83 \( 1 - 116.T + 5.71e5T^{2} \)
89 \( 1 - 813.T + 7.04e5T^{2} \)
97 \( 1 - 794.T + 9.12e5T^{2} \)
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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

−12.66757292263409708323066742514, −11.35235431967553903999591595823, −10.31819639495361563011258468862, −9.415087799851179664143929882773, −8.396212568951396220914393036712, −7.52411789311406541898030271039, −5.96520668290534900983468579294, −4.64229333094379753346661223765, −3.25105324535129063392862491881, −0.888174854581830993379655277345, 0.888174854581830993379655277345, 3.25105324535129063392862491881, 4.64229333094379753346661223765, 5.96520668290534900983468579294, 7.52411789311406541898030271039, 8.396212568951396220914393036712, 9.415087799851179664143929882773, 10.31819639495361563011258468862, 11.35235431967553903999591595823, 12.66757292263409708323066742514

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