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 − 2·11-s − 13-s − 14-s + 16-s − 3·17-s + 2·22-s + 23-s + 26-s + 28-s + 5·29-s + 7·31-s − 32-s + 3·34-s − 2·37-s − 7·41-s − 11·43-s − 2·44-s − 46-s − 8·47-s + 49-s − 52-s + 53-s − 56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.603·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.426·22-s + 0.208·23-s + 0.196·26-s + 0.188·28-s + 0.928·29-s + 1.25·31-s − 0.176·32-s + 0.514·34-s − 0.328·37-s − 1.09·41-s − 1.67·43-s − 0.301·44-s − 0.147·46-s − 1.16·47-s + 1/7·49-s − 0.138·52-s + 0.137·53-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  =  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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.366632324211996219991005794548, −7.73331032905174458096636375950, −6.85137733747305961108725942830, −6.29178096130104433835567544947, −5.15483028346137606042980515566, −4.58607310529684007211324334325, −3.29455399221364617927893806715, −2.44116661444188442116718256018, −1.41570341834852844853092885995, 0, 1.41570341834852844853092885995, 2.44116661444188442116718256018, 3.29455399221364617927893806715, 4.58607310529684007211324334325, 5.15483028346137606042980515566, 6.29178096130104433835567544947, 6.85137733747305961108725942830, 7.73331032905174458096636375950, 8.366632324211996219991005794548

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