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 + 12-s − 4·13-s − 4·14-s + 16-s + 6·17-s + 18-s + 19-s − 4·21-s − 6·23-s + 24-s − 5·25-s − 4·26-s + 27-s − 4·28-s + 6·29-s + 2·31-s + 32-s + 6·34-s + 36-s − 4·37-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 + 0.288·12-s − 1.10·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.229·19-s − 0.872·21-s − 1.25·23-s + 0.204·24-s − 25-s − 0.784·26-s + 0.192·27-s − 0.755·28-s + 1.11·29-s + 0.359·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.657·37-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  =  $(2,\ 114,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.574435356$
$L(\frac12)$  $\approx$  $1.574435356$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
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

−19.96401503206313, −19.54970001118208, −18.67062952744394, −17.20471455464614, −16.17767232529587, −15.55211819207346, −14.35967218034602, −13.70443556114571, −12.53921727142600, −12.03323217483777, −10.14217150491025, −9.659160529040684, −7.973335490421157, −6.847322338295428, −5.617192832220034, −3.896745031646374, −2.722514845444523, 2.722514845444523, 3.896745031646374, 5.617192832220034, 6.847322338295428, 7.973335490421157, 9.659160529040684, 10.14217150491025, 12.03323217483777, 12.53921727142600, 13.70443556114571, 14.35967218034602, 15.55211819207346, 16.17767232529587, 17.20471455464614, 18.67062952744394, 19.54970001118208, 19.96401503206313

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