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  − 1.80·2-s − 0.445·3-s + 2.24·4-s + 1.24·5-s + 0.801·6-s − 2.24·8-s − 0.801·9-s − 2.24·10-s − 12-s − 0.554·15-s + 1.80·16-s + 1.44·18-s − 1.80·19-s + 2.80·20-s + 1.00·24-s + 0.554·25-s + 0.801·27-s − 0.445·29-s + 0.999·30-s − 1.00·32-s − 1.80·36-s − 1.80·37-s + 3.24·38-s − 2.80·40-s + 1.24·43-s − 45-s − 0.801·48-s + ⋯
L(s)  = 1  − 1.80·2-s − 0.445·3-s + 2.24·4-s + 1.24·5-s + 0.801·6-s − 2.24·8-s − 0.801·9-s − 2.24·10-s − 12-s − 0.554·15-s + 1.80·16-s + 1.44·18-s − 1.80·19-s + 2.80·20-s + 1.00·24-s + 0.554·25-s + 0.801·27-s − 0.445·29-s + 0.999·30-s − 1.00·32-s − 1.80·36-s − 1.80·37-s + 3.24·38-s − 2.80·40-s + 1.24·43-s − 45-s − 0.801·48-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  =  $(2,\ 71,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.2523313857$
$L(\frac12)$  $\approx$  $0.2523313857$
$L(1)$   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 \( 1 + 1.80T + T^{2} \)
3 \( 1 + 0.445T + T^{2} \)
5 \( 1 - 1.24T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.80T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 0.445T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.80T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.24T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
73 \( 1 - 1.24T + T^{2} \)
79 \( 1 - 1.24T + T^{2} \)
83 \( 1 + 1.80T + T^{2} \)
89 \( 1 + 0.445T + T^{2} \)
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

−15.30626946792415563634729611589, −13.98183065633101690550037886908, −12.42351194541329992426458232503, −11.05204135964904542931077288526, −10.37263061263202496602275828991, −9.249596659145116242975209696115, −8.392664296558968410521780245040, −6.76877928720481535930982673330, −5.76082408647750230290643880271, −2.19448912992332732423794638884, 2.19448912992332732423794638884, 5.76082408647750230290643880271, 6.76877928720481535930982673330, 8.392664296558968410521780245040, 9.249596659145116242975209696115, 10.37263061263202496602275828991, 11.05204135964904542931077288526, 12.42351194541329992426458232503, 13.98183065633101690550037886908, 15.30626946792415563634729611589

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