Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 79 \cdot 211 $
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 + 2·7-s − 8-s + 9-s + 2·10-s − 2·11-s − 12-s + 5·13-s − 2·14-s + 2·15-s + 16-s + 3·17-s − 18-s + 6·19-s − 2·20-s − 2·21-s + 2·22-s − 23-s + 24-s − 25-s − 5·26-s − 27-s + 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.603·11-s − 0.288·12-s + 1.38·13-s − 0.534·14-s + 0.516·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 1.37·19-s − 0.447·20-s − 0.436·21-s + 0.426·22-s − 0.208·23-s + 0.204·24-s − 1/5·25-s − 0.980·26-s − 0.192·27-s + 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100014\)    =    \(2 \cdot 3 \cdot 79 \cdot 211\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{100014} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 100014,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.297654055$
$L(\frac12)$  $\approx$  $1.297654055$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;79,\;211\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;79,\;211\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
79 \( 1 + T \)
211 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−13.72310627146504, −13.30756962422663, −12.56066825344374, −12.07592247556092, −11.64710679649461, −11.31633091708068, −10.93805064094585, −10.31172354533093, −9.925612632177793, −9.316202282754504, −8.664798875779035, −8.046771759867605, −7.903562550685243, −7.455384197969474, −6.738694470279031, −6.137608628668636, −5.658431264576256, −5.101896948635210, −4.421808480578450, −3.922122438458425, −3.156878533605651, −2.715767671316216, −1.543284218806063, −1.206006633370843, −0.4753763580722994, 0.4753763580722994, 1.206006633370843, 1.543284218806063, 2.715767671316216, 3.156878533605651, 3.922122438458425, 4.421808480578450, 5.101896948635210, 5.658431264576256, 6.137608628668636, 6.738694470279031, 7.455384197969474, 7.903562550685243, 8.046771759867605, 8.664798875779035, 9.316202282754504, 9.925612632177793, 10.31172354533093, 10.93805064094585, 11.31633091708068, 11.64710679649461, 12.07592247556092, 12.56066825344374, 13.30756962422663, 13.72310627146504

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