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 + 4.89·11-s − 4.44·13-s − 14-s + 16-s + 2·17-s + 1.55·19-s − 4.89·22-s + 2.89·23-s + 4.44·26-s + 28-s − 6.89·29-s + 8.89·31-s − 32-s − 2·34-s − 2·37-s − 1.55·38-s + 1.10·41-s + 0.898·43-s + 4.89·44-s − 2.89·46-s + 8.89·47-s + 49-s − 4.44·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.377·7-s − 0.353·8-s + 1.47·11-s − 1.23·13-s − 0.267·14-s + 0.250·16-s + 0.485·17-s + 0.355·19-s − 1.04·22-s + 0.604·23-s + 0.872·26-s + 0.188·28-s − 1.28·29-s + 1.59·31-s − 0.176·32-s − 0.342·34-s − 0.328·37-s − 0.251·38-s + 0.171·41-s + 0.137·43-s + 0.738·44-s − 0.427·46-s + 1.29·47-s + 0.142·49-s − 0.617·52-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  =  $(2,\ 3150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.480401684$
$L(\frac12)$  $\approx$  $1.480401684$
$L(\frac{3}{2})$   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 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 + 4.44T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 1.55T + 19T^{2} \)
23 \( 1 - 2.89T + 23T^{2} \)
29 \( 1 + 6.89T + 29T^{2} \)
31 \( 1 - 8.89T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 1.10T + 41T^{2} \)
43 \( 1 - 0.898T + 43T^{2} \)
47 \( 1 - 8.89T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 - 1.55T + 59T^{2} \)
61 \( 1 - 3.55T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 - 1.10T + 71T^{2} \)
73 \( 1 + 2.89T + 73T^{2} \)
79 \( 1 - 6.89T + 79T^{2} \)
83 \( 1 + 2.44T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
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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

−8.748720080240834891533252527474, −7.945031765864917522280222777612, −7.25302146126156984086378209602, −6.65841241549936900798256318312, −5.73838245348089955810376592647, −4.85373147924713434409810021807, −3.93562728597974423781216070489, −2.91011075334615027338934830526, −1.85788147732680409848894070100, −0.847387495208084136665961882814, 0.847387495208084136665961882814, 1.85788147732680409848894070100, 2.91011075334615027338934830526, 3.93562728597974423781216070489, 4.85373147924713434409810021807, 5.73838245348089955810376592647, 6.65841241549936900798256318312, 7.25302146126156984086378209602, 7.945031765864917522280222777612, 8.748720080240834891533252527474

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