Properties

Label 2-154-1.1-c1-0-3
Degree $2$
Conductor $154$
Sign $1$
Analytic cond. $1.22969$
Root an. cond. $1.10891$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 1.23·3-s + 4-s − 1.23·5-s + 1.23·6-s + 7-s + 8-s − 1.47·9-s − 1.23·10-s + 11-s + 1.23·12-s − 3.23·13-s + 14-s − 1.52·15-s + 16-s + 2.47·17-s − 1.47·18-s − 7.23·19-s − 1.23·20-s + 1.23·21-s + 22-s + 4·23-s + 1.23·24-s − 3.47·25-s − 3.23·26-s − 5.52·27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.713·3-s + 0.5·4-s − 0.552·5-s + 0.504·6-s + 0.377·7-s + 0.353·8-s − 0.490·9-s − 0.390·10-s + 0.301·11-s + 0.356·12-s − 0.897·13-s + 0.267·14-s − 0.394·15-s + 0.250·16-s + 0.599·17-s − 0.346·18-s − 1.66·19-s − 0.276·20-s + 0.269·21-s + 0.213·22-s + 0.834·23-s + 0.252·24-s − 0.694·25-s − 0.634·26-s − 1.06·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(1.22969\)
Root analytic conductor: \(1.10891\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.803645298\)
\(L(\frac12)\) \(\approx\) \(1.803645298\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
good3 \( 1 - 1.23T + 3T^{2} \)
5 \( 1 + 1.23T + 5T^{2} \)
13 \( 1 + 3.23T + 13T^{2} \)
17 \( 1 - 2.47T + 17T^{2} \)
19 \( 1 + 7.23T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 6.94T + 37T^{2} \)
41 \( 1 + 2.47T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 - 8.47T + 53T^{2} \)
59 \( 1 - 2.76T + 59T^{2} \)
61 \( 1 + 0.763T + 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 - 6.47T + 71T^{2} \)
73 \( 1 - 12.9T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−13.04142939108390237862827135983, −12.02573754642256028029402201417, −11.24696787726035266900578757471, −9.971163456632217281631794963600, −8.595221279802153733551270833364, −7.80396324611489422793705431355, −6.52527085313281038885199593615, −5.03379623435561269802443819410, −3.80599127239261113785174017657, −2.44971458371973372755039334054, 2.44971458371973372755039334054, 3.80599127239261113785174017657, 5.03379623435561269802443819410, 6.52527085313281038885199593615, 7.80396324611489422793705431355, 8.595221279802153733551270833364, 9.971163456632217281631794963600, 11.24696787726035266900578757471, 12.02573754642256028029402201417, 13.04142939108390237862827135983

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