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 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 7-s − 8-s − 13-s − 14-s + 16-s − 3·17-s + 2·19-s − 3·23-s + 26-s + 28-s − 3·29-s − 31-s − 32-s + 3·34-s + 2·37-s − 2·38-s + 3·41-s − 7·43-s + 3·46-s − 6·47-s + 49-s − 52-s − 9·53-s − 56-s + 3·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.458·19-s − 0.625·23-s + 0.196·26-s + 0.188·28-s − 0.557·29-s − 0.179·31-s − 0.176·32-s + 0.514·34-s + 0.328·37-s − 0.324·38-s + 0.468·41-s − 1.06·43-s + 0.442·46-s − 0.875·47-s + 1/7·49-s − 0.138·52-s − 1.23·53-s − 0.133·56-s + 0.393·58-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  =  1
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−8.257400357303861005923811617413, −7.72552617683478337861731682443, −6.92510558510511886134207168618, −6.19467630279062369987677517167, −5.29335648956038306490960479982, −4.44000674882418286819423956557, −3.40371943274099612848009659039, −2.35705124590625994497175713453, −1.45137334552753847373459177219, 0, 1.45137334552753847373459177219, 2.35705124590625994497175713453, 3.40371943274099612848009659039, 4.44000674882418286819423956557, 5.29335648956038306490960479982, 6.19467630279062369987677517167, 6.92510558510511886134207168618, 7.72552617683478337861731682443, 8.257400357303861005923811617413

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