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 + 2·11-s + 7·13-s − 14-s + 16-s + 7·17-s + 8·19-s − 2·22-s + 5·23-s − 7·26-s + 28-s − 9·29-s + 31-s − 32-s − 7·34-s − 2·37-s − 8·38-s − 11·41-s + 3·43-s + 2·44-s − 5·46-s − 4·47-s + 49-s + 7·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.603·11-s + 1.94·13-s − 0.267·14-s + 1/4·16-s + 1.69·17-s + 1.83·19-s − 0.426·22-s + 1.04·23-s − 1.37·26-s + 0.188·28-s − 1.67·29-s + 0.179·31-s − 0.176·32-s − 1.20·34-s − 0.328·37-s − 1.29·38-s − 1.71·41-s + 0.457·43-s + 0.301·44-s − 0.737·46-s − 0.583·47-s + 1/7·49-s + 0.970·52-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  =  $(2,\ 3150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.853888543$
$L(\frac12)$  $\approx$  $1.853888543$
$L(\frac{3}{2})$   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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−8.821648821316085235113260169228, −7.86087439724227841038035085522, −7.46903630537801045214345517800, −6.47201154785901138794964613757, −5.73443884214178855015116483866, −5.03532957760104748285298394108, −3.52399453372567038953215375137, −3.32354251010430335657934626486, −1.57259310028887997451852526697, −1.06785029157502222685241757336, 1.06785029157502222685241757336, 1.57259310028887997451852526697, 3.32354251010430335657934626486, 3.52399453372567038953215375137, 5.03532957760104748285298394108, 5.73443884214178855015116483866, 6.47201154785901138794964613757, 7.46903630537801045214345517800, 7.86087439724227841038035085522, 8.821648821316085235113260169228

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