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  + 1.24·2-s − 1.80·3-s + 0.554·4-s − 0.445·5-s − 2.24·6-s − 0.554·8-s + 2.24·9-s − 0.554·10-s − 0.999·12-s + 0.801·15-s − 1.24·16-s + 2.80·18-s + 1.24·19-s − 0.246·20-s + 1.00·24-s − 0.801·25-s − 2.24·27-s − 1.80·29-s + 0.999·30-s − 0.999·32-s + 1.24·36-s + 1.24·37-s + 1.55·38-s + 0.246·40-s − 0.445·43-s − 45-s + 2.24·48-s + ⋯
L(s)  = 1  + 1.24·2-s − 1.80·3-s + 0.554·4-s − 0.445·5-s − 2.24·6-s − 0.554·8-s + 2.24·9-s − 0.554·10-s − 0.999·12-s + 0.801·15-s − 1.24·16-s + 2.80·18-s + 1.24·19-s − 0.246·20-s + 1.00·24-s − 0.801·25-s − 2.24·27-s − 1.80·29-s + 0.999·30-s − 0.999·32-s + 1.24·36-s + 1.24·37-s + 1.55·38-s + 0.246·40-s − 0.445·43-s − 45-s + 2.24·48-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  =  $(2,\ 71,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.5214110656$
$L(\frac12)$  $\approx$  $0.5214110656$
$L(1)$   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 \( 1 - 1.24T + T^{2} \)
3 \( 1 + 1.80T + T^{2} \)
5 \( 1 + 0.445T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.24T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.80T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.24T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 0.445T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
73 \( 1 + 0.445T + T^{2} \)
79 \( 1 + 0.445T + T^{2} \)
83 \( 1 - 1.24T + T^{2} \)
89 \( 1 + 1.80T + T^{2} \)
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

−15.04069640003003501555800810731, −13.58399519906890803499351778176, −12.62842911673974380337261507666, −11.75493220562099759167484344236, −11.17081547941659221637655464575, −9.646341880691708918339787590058, −7.34866153009380292102541584258, −6.02203245587172561884189386059, −5.19361966854786324344583978535, −3.95531641366854447656333489742, 3.95531641366854447656333489742, 5.19361966854786324344583978535, 6.02203245587172561884189386059, 7.34866153009380292102541584258, 9.646341880691708918339787590058, 11.17081547941659221637655464575, 11.75493220562099759167484344236, 12.62842911673974380337261507666, 13.58399519906890803499351778176, 15.04069640003003501555800810731

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