Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s − 2·7-s + 8-s + 9-s − 10-s + 12-s − 4·13-s − 2·14-s − 15-s + 16-s − 8·17-s + 18-s − 19-s − 20-s − 2·21-s + 24-s − 4·25-s − 4·26-s + 27-s − 2·28-s − 6·29-s − 30-s − 9·31-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 1.10·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 1.94·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.436·21-s + 0.204·24-s − 4/5·25-s − 0.784·26-s + 0.192·27-s − 0.377·28-s − 1.11·29-s − 0.182·30-s − 1.61·31-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  =  2
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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
79 \( 1 - T \)
211 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.24977320092718, −13.75301847663385, −13.11640552204659, −12.85951219303186, −12.58115696215251, −11.84702610543463, −11.29068187116155, −11.00081523480276, −10.38335864849867, −9.584580513885328, −9.407988172741980, −8.915290399137753, −8.069402991102701, −7.698343033519924, −7.220533756775885, −6.533147393217878, −6.353172562708772, −5.501671892970491, −4.825327558885636, −4.460236772414722, −3.774960913166492, −3.383599059724343, −2.706293158289772, −2.083752620191814, −1.654750477135382, 0, 0, 1.654750477135382, 2.083752620191814, 2.706293158289772, 3.383599059724343, 3.774960913166492, 4.460236772414722, 4.825327558885636, 5.501671892970491, 6.353172562708772, 6.533147393217878, 7.220533756775885, 7.698343033519924, 8.069402991102701, 8.915290399137753, 9.407988172741980, 9.584580513885328, 10.38335864849867, 11.00081523480276, 11.29068187116155, 11.84702610543463, 12.58115696215251, 12.85951219303186, 13.11640552204659, 13.75301847663385, 14.24977320092718

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