Properties

Degree 2
Conductor $ 2 \cdot 59 $
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 + 2·3-s + 4-s + 2·5-s − 2·6-s − 3·7-s − 8-s + 9-s − 2·10-s + 11-s + 2·12-s + 3·13-s + 3·14-s + 4·15-s + 16-s − 17-s − 18-s − 8·19-s + 2·20-s − 6·21-s − 22-s + 8·23-s − 2·24-s − 25-s − 3·26-s − 4·27-s − 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s − 0.816·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.577·12-s + 0.832·13-s + 0.801·14-s + 1.03·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.83·19-s + 0.447·20-s − 1.30·21-s − 0.213·22-s + 1.66·23-s − 0.408·24-s − 1/5·25-s − 0.588·26-s − 0.769·27-s − 0.566·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 118 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 118 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(118\)    =    \(2 \cdot 59\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{118} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 118,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.099007747$
$L(\frac12)$  $\approx$  $1.099007747$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;59\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;59\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
59 \( 1 - T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−19.82131448179674, −19.25129954146009, −18.48940370995662, −17.26297268609909, −16.59208394504357, −15.36200504062162, −14.61576429167843, −13.35728386235134, −12.92614785249208, −11.13280680579321, −10.02912977105635, −9.097863835607900, −8.593369678016383, −6.997124952982002, −5.982211173926604, −3.562835308056955, −2.199392974311992, 2.199392974311992, 3.562835308056955, 5.982211173926604, 6.997124952982002, 8.593369678016383, 9.097863835607900, 10.02912977105635, 11.13280680579321, 12.92614785249208, 13.35728386235134, 14.61576429167843, 15.36200504062162, 16.59208394504357, 17.26297268609909, 18.48940370995662, 19.25129954146009, 19.82131448179674

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