Properties

Label 2-294-1.1-c3-0-18
Degree $2$
Conductor $294$
Sign $-1$
Analytic cond. $17.3465$
Root an. cond. $4.16492$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 4·4-s − 7.41·5-s − 6·6-s + 8·8-s + 9·9-s − 14.8·10-s + 10.4·11-s − 12·12-s − 2.78·13-s + 22.2·15-s + 16·16-s − 50.4·17-s + 18·18-s − 125.·19-s − 29.6·20-s + 20.9·22-s − 182.·23-s − 24·24-s − 70.0·25-s − 5.57·26-s − 27·27-s + 156.·29-s + 44.4·30-s − 139.·31-s + 32·32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.663·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.468·10-s + 0.287·11-s − 0.288·12-s − 0.0594·13-s + 0.382·15-s + 0.250·16-s − 0.719·17-s + 0.235·18-s − 1.50·19-s − 0.331·20-s + 0.203·22-s − 1.65·23-s − 0.204·24-s − 0.560·25-s − 0.0420·26-s − 0.192·27-s + 0.999·29-s + 0.270·30-s − 0.808·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(17.3465\)
Root analytic conductor: \(4.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 294,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 + 3T \)
7 \( 1 \)
good5 \( 1 + 7.41T + 125T^{2} \)
11 \( 1 - 10.4T + 1.33e3T^{2} \)
13 \( 1 + 2.78T + 2.19e3T^{2} \)
17 \( 1 + 50.4T + 4.91e3T^{2} \)
19 \( 1 + 125.T + 6.85e3T^{2} \)
23 \( 1 + 182.T + 1.21e4T^{2} \)
29 \( 1 - 156.T + 2.43e4T^{2} \)
31 \( 1 + 139.T + 2.97e4T^{2} \)
37 \( 1 + 394.T + 5.06e4T^{2} \)
41 \( 1 + 197.T + 6.89e4T^{2} \)
43 \( 1 - 343.T + 7.95e4T^{2} \)
47 \( 1 - 610.T + 1.03e5T^{2} \)
53 \( 1 + 137.T + 1.48e5T^{2} \)
59 \( 1 + 589.T + 2.05e5T^{2} \)
61 \( 1 + 247.T + 2.26e5T^{2} \)
67 \( 1 + 395.T + 3.00e5T^{2} \)
71 \( 1 - 285.T + 3.57e5T^{2} \)
73 \( 1 - 997.T + 3.89e5T^{2} \)
79 \( 1 + 848.T + 4.93e5T^{2} \)
83 \( 1 - 210.T + 5.71e5T^{2} \)
89 \( 1 - 553.T + 7.04e5T^{2} \)
97 \( 1 - 903.T + 9.12e5T^{2} \)
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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

−10.98490610179967236569511807936, −10.33547488382100185905741210456, −8.881482788131080571472108435132, −7.77898897740144004195363965895, −6.70277185182823508289829043185, −5.86472909891021527139150189748, −4.54233451854330371133894481835, −3.80359878757720019177750551303, −2.04672904305431429835557206459, 0, 2.04672904305431429835557206459, 3.80359878757720019177750551303, 4.54233451854330371133894481835, 5.86472909891021527139150189748, 6.70277185182823508289829043185, 7.77898897740144004195363965895, 8.881482788131080571472108435132, 10.33547488382100185905741210456, 10.98490610179967236569511807936

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