Properties

Degree 2
Conductor $ 2 \cdot 89 $
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 + 3-s + 4-s + 3·5-s + 6-s − 4·7-s + 8-s − 2·9-s + 3·10-s − 6·11-s + 12-s + 2·13-s − 4·14-s + 3·15-s + 16-s + 3·17-s − 2·18-s + 5·19-s + 3·20-s − 4·21-s − 6·22-s − 3·23-s + 24-s + 4·25-s + 2·26-s − 5·27-s − 4·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s − 2/3·9-s + 0.948·10-s − 1.80·11-s + 0.288·12-s + 0.554·13-s − 1.06·14-s + 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.471·18-s + 1.14·19-s + 0.670·20-s − 0.872·21-s − 1.27·22-s − 0.625·23-s + 0.204·24-s + 4/5·25-s + 0.392·26-s − 0.962·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 178 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(178\)    =    \(2 \cdot 89\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{178} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 178,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.968033122$
$L(\frac12)$  $\approx$  $1.968033122$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;89\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;89\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
89 \( 1 + T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−18.99029403327895, −18.19343214131920, −17.10275668620143, −16.06235388508060, −15.53018605857996, −14.12976205190362, −13.58924023539730, −13.09971700535401, −12.01521975837501, −10.38183771626814, −9.924983367076293, −8.721731410942194, −7.364106897454333, −5.947440230223065, −5.481978811610622, −3.315343923846180, −2.519942691765624, 2.519942691765624, 3.315343923846180, 5.481978811610622, 5.947440230223065, 7.364106897454333, 8.721731410942194, 9.924983367076293, 10.38183771626814, 12.01521975837501, 13.09971700535401, 13.58924023539730, 14.12976205190362, 15.53018605857996, 16.06235388508060, 17.10275668620143, 18.19343214131920, 18.99029403327895

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