Properties

Label 2-294-1.1-c3-0-19
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 − 4.58·5-s − 6·6-s + 8·8-s + 9·9-s − 9.17·10-s − 6.48·11-s − 12·12-s − 45.2·13-s + 13.7·15-s + 16·16-s − 81.5·17-s + 18·18-s + 5.05·19-s − 18.3·20-s − 12.9·22-s + 106.·23-s − 24·24-s − 103.·25-s − 90.4·26-s − 27·27-s − 268.·29-s + 27.5·30-s − 292.·31-s + 32·32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.410·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.290·10-s − 0.177·11-s − 0.288·12-s − 0.964·13-s + 0.236·15-s + 0.250·16-s − 1.16·17-s + 0.235·18-s + 0.0610·19-s − 0.205·20-s − 0.125·22-s + 0.963·23-s − 0.204·24-s − 0.831·25-s − 0.682·26-s − 0.192·27-s − 1.71·29-s + 0.167·30-s − 1.69·31-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 \( 1 + 4.58T + 125T^{2} \)
11 \( 1 + 6.48T + 1.33e3T^{2} \)
13 \( 1 + 45.2T + 2.19e3T^{2} \)
17 \( 1 + 81.5T + 4.91e3T^{2} \)
19 \( 1 - 5.05T + 6.85e3T^{2} \)
23 \( 1 - 106.T + 1.21e4T^{2} \)
29 \( 1 + 268.T + 2.43e4T^{2} \)
31 \( 1 + 292.T + 2.97e4T^{2} \)
37 \( 1 - 114.T + 5.06e4T^{2} \)
41 \( 1 - 161.T + 6.89e4T^{2} \)
43 \( 1 + 471.T + 7.95e4T^{2} \)
47 \( 1 + 346.T + 1.03e5T^{2} \)
53 \( 1 - 405.T + 1.48e5T^{2} \)
59 \( 1 - 253.T + 2.05e5T^{2} \)
61 \( 1 - 751.T + 2.26e5T^{2} \)
67 \( 1 - 11.6T + 3.00e5T^{2} \)
71 \( 1 + 681.T + 3.57e5T^{2} \)
73 \( 1 + 685.T + 3.89e5T^{2} \)
79 \( 1 - 0.264T + 4.93e5T^{2} \)
83 \( 1 - 437.T + 5.71e5T^{2} \)
89 \( 1 - 58.5T + 7.04e5T^{2} \)
97 \( 1 - 1.28e3T + 9.12e5T^{2} \)
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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

−11.27890078967850295722266926844, −10.14930693512627050254379731050, −9.036961367638614178984677172686, −7.59734249548219087572375146894, −6.90249757990508154268726875831, −5.65628817233425873590485737802, −4.77098808881376460145599275815, −3.65716026932862719939478241863, −2.08297972204198610759983918240, 0, 2.08297972204198610759983918240, 3.65716026932862719939478241863, 4.77098808881376460145599275815, 5.65628817233425873590485737802, 6.90249757990508154268726875831, 7.59734249548219087572375146894, 9.036961367638614178984677172686, 10.14930693512627050254379731050, 11.27890078967850295722266926844

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