Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 13 $
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 + 2·5-s + 6-s + 4·7-s − 8-s + 9-s − 2·10-s − 4·11-s − 12-s + 13-s − 4·14-s − 2·15-s + 16-s + 2·17-s − 18-s − 8·19-s + 2·20-s − 4·21-s + 4·22-s + 24-s − 25-s − 26-s − 27-s + 4·28-s + 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.288·12-s + 0.277·13-s − 1.06·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.83·19-s + 0.447·20-s − 0.872·21-s + 0.852·22-s + 0.204·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.755·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(78\)    =    \(2 \cdot 3 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{78} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 78,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7252179630$
$L(\frac12)$  $\approx$  $0.7252179630$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 + T \)
13 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−19.07652715819937, −18.16006401535164, −17.58142440802291, −16.87526462306224, −15.60885251069688, −14.56548660784257, −13.34702172357781, −12.08425672990514, −10.80469174604554, −10.33346044923143, −8.724491555695014, −7.702236503545356, −6.106649187507574, −4.916863298177852, −1.961456787437447, 1.961456787437447, 4.916863298177852, 6.106649187507574, 7.702236503545356, 8.724491555695014, 10.33346044923143, 10.80469174604554, 12.08425672990514, 13.34702172357781, 14.56548660784257, 15.60885251069688, 16.87526462306224, 17.58142440802291, 18.16006401535164, 19.07652715819937

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