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 − 3-s + 4-s + 5-s − 6-s + 3·7-s + 8-s − 2·9-s + 10-s + 2·11-s − 12-s − 6·13-s + 3·14-s − 15-s + 16-s − 2·17-s − 2·18-s − 5·19-s + 20-s − 3·21-s + 2·22-s + 4·23-s − 24-s − 4·25-s − 6·26-s + 5·27-s + 3·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s − 1.66·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.471·18-s − 1.14·19-s + 0.223·20-s − 0.654·21-s + 0.426·22-s + 0.834·23-s − 0.204·24-s − 4/5·25-s − 1.17·26-s + 0.962·27-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  =  $(2,\ 118,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.395380631$
$L(\frac12)$  $\approx$  $1.395380631$
$L(\frac{3}{2})$   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 \( 1 + T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−19.40074049884561, −17.92140012403752, −17.17192276604117, −16.73173870359714, −14.92219429967568, −14.71136111495298, −13.56490620862584, −12.38291630889487, −11.53417268299930, −10.79311227912828, −9.366123759317716, −7.930773369176095, −6.589913161686061, −5.414025377597769, −4.472762287466208, −2.288061772225737, 2.288061772225737, 4.472762287466208, 5.414025377597769, 6.589913161686061, 7.930773369176095, 9.366123759317716, 10.79311227912828, 11.53417268299930, 12.38291630889487, 13.56490620862584, 14.71136111495298, 14.92219429967568, 16.73173870359714, 17.17192276604117, 17.92140012403752, 19.40074049884561

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