Properties

Label 2-154-1.1-c1-0-0
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 − 3.23·3-s + 4-s + 3.23·5-s − 3.23·6-s + 7-s + 8-s + 7.47·9-s + 3.23·10-s + 11-s − 3.23·12-s + 1.23·13-s + 14-s − 10.4·15-s + 16-s − 6.47·17-s + 7.47·18-s − 2.76·19-s + 3.23·20-s − 3.23·21-s + 22-s + 4·23-s − 3.23·24-s + 5.47·25-s + 1.23·26-s − 14.4·27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.86·3-s + 0.5·4-s + 1.44·5-s − 1.32·6-s + 0.377·7-s + 0.353·8-s + 2.49·9-s + 1.02·10-s + 0.301·11-s − 0.934·12-s + 0.342·13-s + 0.267·14-s − 2.70·15-s + 0.250·16-s − 1.56·17-s + 1.76·18-s − 0.634·19-s + 0.723·20-s − 0.706·21-s + 0.213·22-s + 0.834·23-s − 0.660·24-s + 1.09·25-s + 0.242·26-s − 2.78·27-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

\(L(1)\) \(\approx\) \(1.246811670\)
\(L(\frac12)\) \(\approx\) \(1.246811670\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.23T + 3T^{2} \)
5 \( 1 - 3.23T + 5T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 + 6.47T + 17T^{2} \)
19 \( 1 + 2.76T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 - 6.47T + 41T^{2} \)
43 \( 1 + 1.52T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 + 0.472T + 53T^{2} \)
59 \( 1 - 7.23T + 59T^{2} \)
61 \( 1 + 5.23T + 61T^{2} \)
67 \( 1 + 15.4T + 67T^{2} \)
71 \( 1 + 2.47T + 71T^{2} \)
73 \( 1 + 4.94T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 3.52T + 97T^{2} \)
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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

−13.02165373672923789671526488169, −11.95456775219894293344517668118, −10.97803505558155713549145959874, −10.46813122479265160040897078207, −9.141839153069202927093459272189, −6.94310403447038207708400147483, −6.23746833597972884543119043828, −5.40100649640324540624234594736, −4.45294812940400001555688902048, −1.77756563632127092242172589873, 1.77756563632127092242172589873, 4.45294812940400001555688902048, 5.40100649640324540624234594736, 6.23746833597972884543119043828, 6.94310403447038207708400147483, 9.141839153069202927093459272189, 10.46813122479265160040897078207, 10.97803505558155713549145959874, 11.95456775219894293344517668118, 13.02165373672923789671526488169

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