Properties

Degree 2
Conductor 71
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.445·2-s + 1.24·3-s − 0.801·4-s − 1.80·5-s − 0.554·6-s + 0.801·8-s + 0.554·9-s + 0.801·10-s − 12-s − 2.24·15-s + 0.445·16-s − 0.246·18-s − 0.445·19-s + 1.44·20-s + 24-s + 2.24·25-s − 0.554·27-s + 1.24·29-s + 0.999·30-s − 32-s − 0.445·36-s − 0.445·37-s + 0.198·38-s − 1.44·40-s − 1.80·43-s − 0.999·45-s + 0.554·48-s + ⋯
L(s)  = 1  − 0.445·2-s + 1.24·3-s − 0.801·4-s − 1.80·5-s − 0.554·6-s + 0.801·8-s + 0.554·9-s + 0.801·10-s − 12-s − 2.24·15-s + 0.445·16-s − 0.246·18-s − 0.445·19-s + 1.44·20-s + 24-s + 2.24·25-s − 0.554·27-s + 1.24·29-s + 0.999·30-s − 32-s − 0.445·36-s − 0.445·37-s + 0.198·38-s − 1.44·40-s − 1.80·43-s − 0.999·45-s + 0.554·48-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(71\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{71} (70, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 71,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.4178985571$
$L(\frac12)$  $\approx$  $0.4178985571$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 71$, \(F_p\) is a polynomial of degree 2. If $p = 71$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad71 \( 1 - T \)
good2 \( 1 + 0.445T + T^{2} \)
3 \( 1 - 1.24T + T^{2} \)
5 \( 1 + 1.80T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 0.445T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 1.24T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 0.445T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.80T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
73 \( 1 + 1.80T + T^{2} \)
79 \( 1 + 1.80T + T^{2} \)
83 \( 1 + 0.445T + T^{2} \)
89 \( 1 - 1.24T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−14.92707055095013819463169451241, −14.03910777988174726171598133235, −12.88778415189161766638407934767, −11.67985104016781996899636064300, −10.24785144415084892602116520791, −8.775621347957351956553642935712, −8.308375254782647564178629435651, −7.32935458632589252277476848657, −4.51844117616935213608968885293, −3.39979849702102685652285544820, 3.39979849702102685652285544820, 4.51844117616935213608968885293, 7.32935458632589252277476848657, 8.308375254782647564178629435651, 8.775621347957351956553642935712, 10.24785144415084892602116520791, 11.67985104016781996899636064300, 12.88778415189161766638407934767, 14.03910777988174726171598133235, 14.92707055095013819463169451241

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