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 − 4·11-s − 3·13-s + 14-s + 16-s + 7·17-s − 6·19-s + 4·22-s + 9·23-s + 3·26-s − 28-s − 3·29-s − 7·31-s − 32-s − 7·34-s − 10·37-s + 6·38-s + 41-s + 13·43-s − 4·44-s − 9·46-s − 2·47-s + 49-s − 3·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.20·11-s − 0.832·13-s + 0.267·14-s + 1/4·16-s + 1.69·17-s − 1.37·19-s + 0.852·22-s + 1.87·23-s + 0.588·26-s − 0.188·28-s − 0.557·29-s − 1.25·31-s − 0.176·32-s − 1.20·34-s − 1.64·37-s + 0.973·38-s + 0.156·41-s + 1.98·43-s − 0.603·44-s − 1.32·46-s − 0.291·47-s + 1/7·49-s − 0.416·52-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  =  $(2,\ 3150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9195732093$
$L(\frac12)$  $\approx$  $0.9195732093$
$L(\frac{3}{2})$   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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 - 13 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−8.781675080601501623768376748522, −7.84335343978744776618511390271, −7.36845273584414996909558838079, −6.66264157947553488246671032551, −5.51657674960247696910072397970, −5.14823767607295882644190048281, −3.77034495441787172485473312012, −2.88780756149487526860144751240, −2.05198583129271445231774373106, −0.62005820358442542220858327088, 0.62005820358442542220858327088, 2.05198583129271445231774373106, 2.88780756149487526860144751240, 3.77034495441787172485473312012, 5.14823767607295882644190048281, 5.51657674960247696910072397970, 6.66264157947553488246671032551, 7.36845273584414996909558838079, 7.84335343978744776618511390271, 8.781675080601501623768376748522

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