Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 79 \cdot 211 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 3·5-s + 6-s + 2·7-s + 8-s + 9-s − 3·10-s − 6·11-s + 12-s − 4·13-s + 2·14-s − 3·15-s + 16-s + 6·17-s + 18-s − 7·19-s − 3·20-s + 2·21-s − 6·22-s + 6·23-s + 24-s + 4·25-s − 4·26-s + 27-s + 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s − 1.80·11-s + 0.288·12-s − 1.10·13-s + 0.534·14-s − 0.774·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.60·19-s − 0.670·20-s + 0.436·21-s − 1.27·22-s + 1.25·23-s + 0.204·24-s + 4/5·25-s − 0.784·26-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  =  1
Selberg data  =  $(2,\ 100014,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 3 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.21737817378158, −13.36573664238831, −13.05585879510360, −12.56760770521303, −12.00935610997405, −11.82502312694921, −11.08081890255276, −10.60779353557452, −10.18567518725909, −9.782544782309382, −8.691658918580470, −8.347169106246166, −7.891309477070742, −7.656002159798229, −7.107370170725611, −6.467812331824721, −5.733634668836331, −4.868395247330762, −4.737322890769904, −4.441881897803899, −3.353865656080764, −3.070778001634544, −2.569394602847438, −1.838209429307028, −0.8826076685529489, 0, 0.8826076685529489, 1.838209429307028, 2.569394602847438, 3.070778001634544, 3.353865656080764, 4.441881897803899, 4.737322890769904, 4.868395247330762, 5.733634668836331, 6.467812331824721, 7.107370170725611, 7.656002159798229, 7.891309477070742, 8.347169106246166, 8.691658918580470, 9.782544782309382, 10.18567518725909, 10.60779353557452, 11.08081890255276, 11.82502312694921, 12.00935610997405, 12.56760770521303, 13.05585879510360, 13.36573664238831, 14.21737817378158

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