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 + 6-s + 4·7-s − 8-s + 9-s + 4·11-s − 12-s − 4·14-s + 16-s − 2·17-s − 18-s + 19-s − 4·21-s − 4·22-s − 2·23-s + 24-s − 5·25-s − 27-s + 4·28-s − 6·29-s + 6·31-s − 32-s − 4·33-s + 2·34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 1.06·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.229·19-s − 0.872·21-s − 0.852·22-s − 0.417·23-s + 0.204·24-s − 25-s − 0.192·27-s + 0.755·28-s − 1.11·29-s + 1.07·31-s − 0.176·32-s − 0.696·33-s + 0.342·34-s + 1/6·36-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  :  $\chi_{114} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 114,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7635720565$
$L(\frac12)$  $\approx$  $0.7635720565$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 + T \)
19 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−19.14219438133610, −18.07712678104046, −17.47220779922100, −16.85627512992211, −15.64064726896678, −14.71418297337340, −13.70872882601071, −12.01124919433446, −11.52937511132746, −10.56257717710159, −9.298664195949932, −8.207396006533562, −7.094370199969288, −5.748361641369155, −4.270181187962730, −1.652545614337292, 1.652545614337292, 4.270181187962730, 5.748361641369155, 7.094370199969288, 8.207396006533562, 9.298664195949932, 10.56257717710159, 11.52937511132746, 12.01124919433446, 13.70872882601071, 14.71418297337340, 15.64064726896678, 16.85627512992211, 17.47220779922100, 18.07712678104046, 19.14219438133610

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