Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 7-s − 8-s − 3·11-s + 2·13-s − 14-s + 16-s − 3·17-s − 7·19-s + 3·22-s − 2·26-s + 28-s + 6·29-s − 4·31-s − 32-s + 3·34-s + 8·37-s + 7·38-s + 9·41-s + 8·43-s − 3·44-s + 6·47-s + 49-s + 2·52-s + 12·53-s − 56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.904·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s − 1.60·19-s + 0.639·22-s − 0.392·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.514·34-s + 1.31·37-s + 1.13·38-s + 1.40·41-s + 1.21·43-s − 0.452·44-s + 0.875·47-s + 1/7·49-s + 0.277·52-s + 1.64·53-s − 0.133·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.148260461$
$L(\frac12)$  $\approx$  $1.148260461$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.843907376404927172594903055109, −7.894666672441482585658026129317, −7.48692188901345043617011736607, −6.36479514579877312969576273814, −5.93066704771418886938831449942, −4.75362850282369654321367926180, −4.05297618114505143896388993114, −2.72666069116306601569300211291, −2.06338408386768518663237304419, −0.70479184427930682602517236736, 0.70479184427930682602517236736, 2.06338408386768518663237304419, 2.72666069116306601569300211291, 4.05297618114505143896388993114, 4.75362850282369654321367926180, 5.93066704771418886938831449942, 6.36479514579877312969576273814, 7.48692188901345043617011736607, 7.894666672441482585658026129317, 8.843907376404927172594903055109

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