Properties

Label 2-294-1.1-c5-0-18
Degree $2$
Conductor $294$
Sign $1$
Analytic cond. $47.1528$
Root an. cond. $6.86679$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 9·3-s + 16·4-s + 75.4·5-s + 36·6-s + 64·8-s + 81·9-s + 301.·10-s − 149.·11-s + 144·12-s − 349.·13-s + 679.·15-s + 256·16-s + 1.14e3·17-s + 324·18-s + 2.79e3·19-s + 1.20e3·20-s − 597.·22-s + 1.81e3·23-s + 576·24-s + 2.57e3·25-s − 1.39e3·26-s + 729·27-s − 759.·29-s + 2.71e3·30-s − 9.03e3·31-s + 1.02e3·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.35·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.954·10-s − 0.372·11-s + 0.288·12-s − 0.573·13-s + 0.779·15-s + 0.250·16-s + 0.964·17-s + 0.235·18-s + 1.77·19-s + 0.675·20-s − 0.263·22-s + 0.715·23-s + 0.204·24-s + 0.823·25-s − 0.405·26-s + 0.192·27-s − 0.167·29-s + 0.551·30-s − 1.68·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(47.1528\)
Root analytic conductor: \(6.86679\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(5.444027272\)
\(L(\frac12)\) \(\approx\) \(5.444027272\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4T \)
3 \( 1 - 9T \)
7 \( 1 \)
good5 \( 1 - 75.4T + 3.12e3T^{2} \)
11 \( 1 + 149.T + 1.61e5T^{2} \)
13 \( 1 + 349.T + 3.71e5T^{2} \)
17 \( 1 - 1.14e3T + 1.41e6T^{2} \)
19 \( 1 - 2.79e3T + 2.47e6T^{2} \)
23 \( 1 - 1.81e3T + 6.43e6T^{2} \)
29 \( 1 + 759.T + 2.05e7T^{2} \)
31 \( 1 + 9.03e3T + 2.86e7T^{2} \)
37 \( 1 - 7.79e3T + 6.93e7T^{2} \)
41 \( 1 + 7.64e3T + 1.15e8T^{2} \)
43 \( 1 - 1.21e4T + 1.47e8T^{2} \)
47 \( 1 + 2.45e4T + 2.29e8T^{2} \)
53 \( 1 - 1.35e4T + 4.18e8T^{2} \)
59 \( 1 - 2.63e4T + 7.14e8T^{2} \)
61 \( 1 + 3.53e4T + 8.44e8T^{2} \)
67 \( 1 - 5.43e4T + 1.35e9T^{2} \)
71 \( 1 + 7.01e4T + 1.80e9T^{2} \)
73 \( 1 - 4.44e4T + 2.07e9T^{2} \)
79 \( 1 - 6.16e4T + 3.07e9T^{2} \)
83 \( 1 - 8.71e4T + 3.93e9T^{2} \)
89 \( 1 + 9.85e4T + 5.58e9T^{2} \)
97 \( 1 + 3.23e4T + 8.58e9T^{2} \)
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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

−10.90188447971699712849803674165, −9.807158247010878719365031544446, −9.373660577129236932390163105811, −7.86212993282366863671818867780, −6.98735982932894800674244015423, −5.66837638939159672045978117534, −5.09481266461612846990856491548, −3.44863478789883915223006110258, −2.48038006936622756390708073092, −1.32330637735012584549227107731, 1.32330637735012584549227107731, 2.48038006936622756390708073092, 3.44863478789883915223006110258, 5.09481266461612846990856491548, 5.66837638939159672045978117534, 6.98735982932894800674244015423, 7.86212993282366863671818867780, 9.373660577129236932390163105811, 9.807158247010878719365031544446, 10.90188447971699712849803674165

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