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Properties

Label 2-296-296.147-c0-0-1
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 − 1.61·3-s + 4-s + 0.618·5-s − 1.61·6-s + 8-s + 1.61·9-s + 0.618·10-s + 0.618·11-s − 1.61·12-s − 1.61·13-s − 1.00·15-s + 16-s + 1.61·18-s + 0.618·20-s + 0.618·22-s − 1.61·23-s − 1.61·24-s − 0.618·25-s − 1.61·26-s − 27-s − 1.61·29-s − 1.00·30-s + 0.618·31-s + 32-s − 1.00·33-s + 1.61·36-s + ⋯
L(s)  = 1  + 2-s − 1.61·3-s + 4-s + 0.618·5-s − 1.61·6-s + 8-s + 1.61·9-s + 0.618·10-s + 0.618·11-s − 1.61·12-s − 1.61·13-s − 1.00·15-s + 16-s + 1.61·18-s + 0.618·20-s + 0.618·22-s − 1.61·23-s − 1.61·24-s − 0.618·25-s − 1.61·26-s − 27-s − 1.61·29-s − 1.00·30-s + 0.618·31-s + 32-s − 1.00·33-s + 1.61·36-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

\(L(\frac{1}{2})\) \(\approx\) \(0.9392512365\)
\(L(\frac12)\) \(\approx\) \(0.9392512365\)
\(L(1)\) 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 \( 1 + 1.61T + T^{2} \)
5 \( 1 - 0.618T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 0.618T + T^{2} \)
13 \( 1 + 1.61T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.61T + T^{2} \)
29 \( 1 + 1.61T + T^{2} \)
31 \( 1 - 0.618T + T^{2} \)
41 \( 1 + 1.61T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 0.618T + T^{2} \)
79 \( 1 + 1.61T + T^{2} \)
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

−11.92878601476356497669243678052, −11.48362480243374569351178297044, −10.29549945510468854334529785748, −9.726083695917808617414382926183, −7.64127581030885880174897304762, −6.65934243594218128434794578267, −5.86144625326211303585270062813, −5.12988544395750539868160378016, −4.07545421380961793180420069303, −2.04488031533664882038543243068, 2.04488031533664882038543243068, 4.07545421380961793180420069303, 5.12988544395750539868160378016, 5.86144625326211303585270062813, 6.65934243594218128434794578267, 7.64127581030885880174897304762, 9.726083695917808617414382926183, 10.29549945510468854334529785748, 11.48362480243374569351178297044, 11.92878601476356497669243678052

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