Properties

Label 2-294-1.1-c3-0-6
Degree $2$
Conductor $294$
Sign $1$
Analytic cond. $17.3465$
Root an. cond. $4.16492$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·3-s + 4·4-s + 8·5-s − 6·6-s − 8·8-s + 9·9-s − 16·10-s + 40·11-s + 12·12-s + 4·13-s + 24·15-s + 16·16-s − 84·17-s − 18·18-s + 148·19-s + 32·20-s − 80·22-s + 84·23-s − 24·24-s − 61·25-s − 8·26-s + 27·27-s + 58·29-s − 48·30-s − 136·31-s − 32·32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.715·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.505·10-s + 1.09·11-s + 0.288·12-s + 0.0853·13-s + 0.413·15-s + 1/4·16-s − 1.19·17-s − 0.235·18-s + 1.78·19-s + 0.357·20-s − 0.775·22-s + 0.761·23-s − 0.204·24-s − 0.487·25-s − 0.0603·26-s + 0.192·27-s + 0.371·29-s − 0.292·30-s − 0.787·31-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: yes
Arithmetic: yes
Character: $\chi_{294} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.006687205\)
\(L(\frac12)\) \(\approx\) \(2.006687205\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
3 \( 1 - p T \)
7 \( 1 \)
good5 \( 1 - 8 T + p^{3} T^{2} \)
11 \( 1 - 40 T + p^{3} T^{2} \)
13 \( 1 - 4 T + p^{3} T^{2} \)
17 \( 1 + 84 T + p^{3} T^{2} \)
19 \( 1 - 148 T + p^{3} T^{2} \)
23 \( 1 - 84 T + p^{3} T^{2} \)
29 \( 1 - 2 p T + p^{3} T^{2} \)
31 \( 1 + 136 T + p^{3} T^{2} \)
37 \( 1 + 6 p T + p^{3} T^{2} \)
41 \( 1 - 420 T + p^{3} T^{2} \)
43 \( 1 + 164 T + p^{3} T^{2} \)
47 \( 1 - 488 T + p^{3} T^{2} \)
53 \( 1 - 478 T + p^{3} T^{2} \)
59 \( 1 - 548 T + p^{3} T^{2} \)
61 \( 1 - 692 T + p^{3} T^{2} \)
67 \( 1 + 908 T + p^{3} T^{2} \)
71 \( 1 + 524 T + p^{3} T^{2} \)
73 \( 1 - 440 T + p^{3} T^{2} \)
79 \( 1 - 1216 T + p^{3} T^{2} \)
83 \( 1 + 684 T + p^{3} T^{2} \)
89 \( 1 - 604 T + p^{3} T^{2} \)
97 \( 1 + 832 T + p^{3} T^{2} \)
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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

−11.22894754926272745903460195339, −10.15224328898191139434742000561, −9.241355611549007854133376466335, −8.855951246216007523497240138527, −7.48147140812442966306247349246, −6.67955519858019177128485458236, −5.46100288531879534106624052515, −3.83953552024101295380785475129, −2.43457001579993789323890589457, −1.17754236925011366554141561200, 1.17754236925011366554141561200, 2.43457001579993789323890589457, 3.83953552024101295380785475129, 5.46100288531879534106624052515, 6.67955519858019177128485458236, 7.48147140812442966306247349246, 8.855951246216007523497240138527, 9.241355611549007854133376466335, 10.15224328898191139434742000561, 11.22894754926272745903460195339

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