Properties

Degree $2$
Conductor $294$
Sign $1$
Motivic weight $5$
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 + 66·5-s + 36·6-s + 64·8-s + 81·9-s + 264·10-s − 60·11-s + 144·12-s + 658·13-s + 594·15-s + 256·16-s + 414·17-s + 324·18-s − 956·19-s + 1.05e3·20-s − 240·22-s + 600·23-s + 576·24-s + 1.23e3·25-s + 2.63e3·26-s + 729·27-s + 5.57e3·29-s + 2.37e3·30-s + 3.59e3·31-s + 1.02e3·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.18·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.834·10-s − 0.149·11-s + 0.288·12-s + 1.07·13-s + 0.681·15-s + 1/4·16-s + 0.347·17-s + 0.235·18-s − 0.607·19-s + 0.590·20-s − 0.105·22-s + 0.236·23-s + 0.204·24-s + 0.393·25-s + 0.763·26-s + 0.192·27-s + 1.23·29-s + 0.481·30-s + 0.671·31-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: $1$
Motivic weight: \(5\)
Character: $\chi_{294} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(5.424389917\)
\(L(\frac12)\) \(\approx\) \(5.424389917\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{2} T \)
3 \( 1 - p^{2} T \)
7 \( 1 \)
good5 \( 1 - 66 T + p^{5} T^{2} \)
11 \( 1 + 60 T + p^{5} T^{2} \)
13 \( 1 - 658 T + p^{5} T^{2} \)
17 \( 1 - 414 T + p^{5} T^{2} \)
19 \( 1 + 956 T + p^{5} T^{2} \)
23 \( 1 - 600 T + p^{5} T^{2} \)
29 \( 1 - 5574 T + p^{5} T^{2} \)
31 \( 1 - 3592 T + p^{5} T^{2} \)
37 \( 1 + 8458 T + p^{5} T^{2} \)
41 \( 1 + 19194 T + p^{5} T^{2} \)
43 \( 1 - 13316 T + p^{5} T^{2} \)
47 \( 1 - 19680 T + p^{5} T^{2} \)
53 \( 1 + 31266 T + p^{5} T^{2} \)
59 \( 1 + 26340 T + p^{5} T^{2} \)
61 \( 1 - 31090 T + p^{5} T^{2} \)
67 \( 1 + 16804 T + p^{5} T^{2} \)
71 \( 1 - 6120 T + p^{5} T^{2} \)
73 \( 1 - 25558 T + p^{5} T^{2} \)
79 \( 1 - 74408 T + p^{5} T^{2} \)
83 \( 1 - 6468 T + p^{5} T^{2} \)
89 \( 1 - 32742 T + p^{5} T^{2} \)
97 \( 1 + 166082 T + p^{5} T^{2} \)
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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

−10.79617552596184611305484289769, −10.11630774462742478324994208019, −9.029555867670381142356202516077, −8.112718798013106437475140949894, −6.73027049925005514796468756958, −5.97063122220731253106996152904, −4.86476246428133923610934764365, −3.55961191213915360099516830949, −2.42785062549074626720126549827, −1.32813200601993022889986711952, 1.32813200601993022889986711952, 2.42785062549074626720126549827, 3.55961191213915360099516830949, 4.86476246428133923610934764365, 5.97063122220731253106996152904, 6.73027049925005514796468756958, 8.112718798013106437475140949894, 9.029555867670381142356202516077, 10.11630774462742478324994208019, 10.79617552596184611305484289769

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