Properties

Degree 2
Conductor $ 2^{2} \cdot 5 \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  + 3-s + 5-s + 7-s − 2·9-s + 3·11-s − 13-s + 15-s − 3·17-s + 2·19-s + 21-s − 6·23-s + 25-s − 5·27-s − 9·29-s + 8·31-s + 3·33-s + 35-s − 10·37-s − 39-s + 2·43-s − 2·45-s − 3·47-s + 49-s − 3·51-s + 3·55-s + 2·57-s + 12·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.277·13-s + 0.258·15-s − 0.727·17-s + 0.458·19-s + 0.218·21-s − 1.25·23-s + 1/5·25-s − 0.962·27-s − 1.67·29-s + 1.43·31-s + 0.522·33-s + 0.169·35-s − 1.64·37-s − 0.160·39-s + 0.304·43-s − 0.298·45-s − 0.437·47-s + 1/7·49-s − 0.420·51-s + 0.404·55-s + 0.264·57-s + 1.56·59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{140} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 140,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.344301542$
$L(\frac12)$  $\approx$  $1.344301542$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−19.72929636996841, −18.80171526332270, −17.60666990978814, −17.16117128021197, −15.92986184745730, −14.82557768013544, −14.14613851866987, −13.43051342969939, −12.06507020707333, −11.23967367108057, −9.891670406720995, −8.980861688416761, −8.046356379591346, −6.670508743930514, −5.400181003867136, −3.800302729067453, −2.145871460100972, 2.145871460100972, 3.800302729067453, 5.400181003867136, 6.670508743930514, 8.046356379591346, 8.980861688416761, 9.891670406720995, 11.23967367108057, 12.06507020707333, 13.43051342969939, 14.14613851866987, 14.82557768013544, 15.92986184745730, 17.16117128021197, 17.60666990978814, 18.80171526332270, 19.72929636996841

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