Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 19 $
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 + 8-s + 9-s + 2·10-s − 4·11-s − 12-s + 2·13-s − 2·15-s + 16-s − 6·17-s + 18-s − 19-s + 2·20-s − 4·22-s − 4·23-s − 24-s − 25-s + 2·26-s − 27-s − 2·29-s − 2·30-s + 4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s − 0.288·12-s + 0.554·13-s − 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.229·19-s + 0.447·20-s − 0.852·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.371·29-s − 0.365·30-s + 0.718·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(114\)    =    \(2 \cdot 3 \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{114} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 114,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.38772$
$L(\frac12)$  $\approx$  $1.38772$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−13.26471264838449267214147905866, −12.97088967028240784250602792338, −11.47795618604630761722336104692, −10.67649349923181634582213825679, −9.600878448141019797397427731260, −7.987570621007900466899230715503, −6.46925034993373182658971924423, −5.66755609194805589043695166783, −4.39790054298019034493919320980, −2.33973473634575001863443152397, 2.33973473634575001863443152397, 4.39790054298019034493919320980, 5.66755609194805589043695166783, 6.46925034993373182658971924423, 7.987570621007900466899230715503, 9.600878448141019797397427731260, 10.67649349923181634582213825679, 11.47795618604630761722336104692, 12.97088967028240784250602792338, 13.26471264838449267214147905866

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