Properties

Degree 2
Conductor $ 2 \cdot 29 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s + 4-s − 3·5-s + 3·6-s − 2·7-s − 8-s + 6·9-s + 3·10-s − 11-s − 3·12-s + 3·13-s + 2·14-s + 9·15-s + 16-s − 4·17-s − 6·18-s − 8·19-s − 3·20-s + 6·21-s + 22-s + 3·24-s + 4·25-s − 3·26-s − 9·27-s − 2·28-s − 29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 1.22·6-s − 0.755·7-s − 0.353·8-s + 2·9-s + 0.948·10-s − 0.301·11-s − 0.866·12-s + 0.832·13-s + 0.534·14-s + 2.32·15-s + 1/4·16-s − 0.970·17-s − 1.41·18-s − 1.83·19-s − 0.670·20-s + 1.30·21-s + 0.213·22-s + 0.612·24-s + 4/5·25-s − 0.588·26-s − 1.73·27-s − 0.377·28-s − 0.185·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(58\)    =    \(2 \cdot 29\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{58} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 58,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.37542430604861, −18.82602958445260, −17.68188315517846, −16.82608394495314, −15.84934678481967, −15.47776366797874, −12.96417955859964, −12.07048630043892, −11.08257082663040, −10.46054334513251, −8.643506374817898, −7.092294954930529, −6.117550642980299, −4.221463272161603, 0, 4.221463272161603, 6.117550642980299, 7.092294954930529, 8.643506374817898, 10.46054334513251, 11.08257082663040, 12.07048630043892, 12.96417955859964, 15.47776366797874, 15.84934678481967, 16.82608394495314, 17.68188315517846, 18.82602958445260, 19.37542430604861

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