Properties

Degree 2
Conductor $ 2 \cdot 61 $
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 − 2·3-s + 4-s + 5-s + 2·6-s − 5·7-s − 8-s + 9-s − 10-s − 3·11-s − 2·12-s − 3·13-s + 5·14-s − 2·15-s + 16-s − 18-s + 20-s + 10·21-s + 3·22-s + 5·23-s + 2·24-s − 4·25-s + 3·26-s + 4·27-s − 5·28-s + 6·29-s + 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s − 0.577·12-s − 0.832·13-s + 1.33·14-s − 0.516·15-s + 1/4·16-s − 0.235·18-s + 0.223·20-s + 2.18·21-s + 0.639·22-s + 1.04·23-s + 0.408·24-s − 4/5·25-s + 0.588·26-s + 0.769·27-s − 0.944·28-s + 1.11·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(122\)    =    \(2 \cdot 61\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{122} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 122,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;61\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;61\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
61 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.27670182220699, −18.60907490280299, −17.46140880136919, −16.99149759719765, −16.07445556073780, −15.43985689286291, −13.68858534982883, −12.65219688253045, −11.93416893925430, −10.48908165651655, −10.06476543076194, −8.921814354406228, −7.172444286350841, −6.312794020948259, −5.283240131179681, −2.917503897181098, 0, 2.917503897181098, 5.283240131179681, 6.312794020948259, 7.172444286350841, 8.921814354406228, 10.06476543076194, 10.48908165651655, 11.93416893925430, 12.65219688253045, 13.68858534982883, 15.43985689286291, 16.07445556073780, 16.99149759719765, 17.46140880136919, 18.60907490280299, 19.27670182220699

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