Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 7^{2} $
Sign $-1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 26·3-s + 16·4-s + 25·5-s − 104·6-s − 64·8-s + 433·9-s − 100·10-s − 768·11-s + 416·12-s + 46·13-s + 650·15-s + 256·16-s − 378·17-s − 1.73e3·18-s − 1.10e3·19-s + 400·20-s + 3.07e3·22-s − 1.98e3·23-s − 1.66e3·24-s + 625·25-s − 184·26-s + 4.94e3·27-s − 5.61e3·29-s − 2.60e3·30-s + 3.98e3·31-s − 1.02e3·32-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.66·3-s + 1/2·4-s + 0.447·5-s − 1.17·6-s − 0.353·8-s + 1.78·9-s − 0.316·10-s − 1.91·11-s + 0.833·12-s + 0.0754·13-s + 0.745·15-s + 1/4·16-s − 0.317·17-s − 1.25·18-s − 0.699·19-s + 0.223·20-s + 1.35·22-s − 0.782·23-s − 0.589·24-s + 1/5·25-s − 0.0533·26-s + 1.30·27-s − 1.23·29-s − 0.527·30-s + 0.745·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(5\)
character  :  $\chi_{490} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 490,\ (\ :5/2),\ -1)$
$L(3)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{7}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + p^{2} T \)
5 \( 1 - p^{2} T \)
7 \( 1 \)
good3 \( 1 - 26 T + p^{5} T^{2} \)
11 \( 1 + 768 T + p^{5} T^{2} \)
13 \( 1 - 46 T + p^{5} T^{2} \)
17 \( 1 + 378 T + p^{5} T^{2} \)
19 \( 1 + 1100 T + p^{5} T^{2} \)
23 \( 1 + 1986 T + p^{5} T^{2} \)
29 \( 1 + 5610 T + p^{5} T^{2} \)
31 \( 1 - 3988 T + p^{5} T^{2} \)
37 \( 1 + 142 T + p^{5} T^{2} \)
41 \( 1 + 1542 T + p^{5} T^{2} \)
43 \( 1 + 5026 T + p^{5} T^{2} \)
47 \( 1 + 24738 T + p^{5} T^{2} \)
53 \( 1 + 14166 T + p^{5} T^{2} \)
59 \( 1 + 28380 T + p^{5} T^{2} \)
61 \( 1 + 5522 T + p^{5} T^{2} \)
67 \( 1 + 24742 T + p^{5} T^{2} \)
71 \( 1 - 42372 T + p^{5} T^{2} \)
73 \( 1 - 52126 T + p^{5} T^{2} \)
79 \( 1 + 39640 T + p^{5} T^{2} \)
83 \( 1 - 59826 T + p^{5} T^{2} \)
89 \( 1 + 57690 T + p^{5} T^{2} \)
97 \( 1 - 144382 T + p^{5} T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.669002666708698652507419629977, −8.713468550513807663689213258966, −8.057515697453117079710094908552, −7.48675208064009779694226986121, −6.22682201762564498965641537121, −4.85725248292182215085872859861, −3.41842085723722962705402837830, −2.47555169600571597567642191654, −1.81267118610655376094072918020, 0, 1.81267118610655376094072918020, 2.47555169600571597567642191654, 3.41842085723722962705402837830, 4.85725248292182215085872859861, 6.22682201762564498965641537121, 7.48675208064009779694226986121, 8.057515697453117079710094908552, 8.713468550513807663689213258966, 9.669002666708698652507419629977

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