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Properties

Label 2-296-296.147-c0-0-3
Degree $2$
Conductor $296$
Sign $1$
Analytic cond. $0.147723$
Root an. cond. $0.384347$
Motivic weight $0$
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 + 0.618·3-s + 4-s − 1.61·5-s + 0.618·6-s + 8-s − 0.618·9-s − 1.61·10-s − 1.61·11-s + 0.618·12-s + 0.618·13-s − 1.00·15-s + 16-s − 0.618·18-s − 1.61·20-s − 1.61·22-s + 0.618·23-s + 0.618·24-s + 1.61·25-s + 0.618·26-s − 27-s + 0.618·29-s − 1.00·30-s − 1.61·31-s + 32-s − 1.00·33-s − 0.618·36-s + ⋯
L(s)  = 1  + 2-s + 0.618·3-s + 4-s − 1.61·5-s + 0.618·6-s + 8-s − 0.618·9-s − 1.61·10-s − 1.61·11-s + 0.618·12-s + 0.618·13-s − 1.00·15-s + 16-s − 0.618·18-s − 1.61·20-s − 1.61·22-s + 0.618·23-s + 0.618·24-s + 1.61·25-s + 0.618·26-s − 27-s + 0.618·29-s − 1.00·30-s − 1.61·31-s + 32-s − 1.00·33-s − 0.618·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $1$
Analytic conductor: \(0.147723\)
Root analytic conductor: \(0.384347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{296} (147, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 296,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.226696605\)
\(L(\frac12)\) \(\approx\) \(1.226696605\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
37 \( 1 - T \)
good3 \( 1 - 0.618T + T^{2} \)
5 \( 1 + 1.61T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 1.61T + T^{2} \)
13 \( 1 - 0.618T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 0.618T + T^{2} \)
29 \( 1 - 0.618T + T^{2} \)
31 \( 1 + 1.61T + T^{2} \)
41 \( 1 - 0.618T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.61T + T^{2} \)
79 \( 1 - 0.618T + T^{2} \)
83 \( 1 - 2T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−12.08662681088524732658498686367, −11.15741192033290888839995195064, −10.64210747422095420115882841363, −8.845151764191126308860180953875, −7.85559919905754596078861043978, −7.41867145639848087708097846353, −5.85065420837689405939476020258, −4.68964727189577893116120082812, −3.56342714014302075545816589071, −2.72220946218321241717739633371, 2.72220946218321241717739633371, 3.56342714014302075545816589071, 4.68964727189577893116120082812, 5.85065420837689405939476020258, 7.41867145639848087708097846353, 7.85559919905754596078861043978, 8.845151764191126308860180953875, 10.64210747422095420115882841363, 11.15741192033290888839995195064, 12.08662681088524732658498686367

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