Properties

Degree 2
Conductor $ 2^{4} \cdot 7 \cdot 19 \cdot 47 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 7-s − 3·9-s + 2·11-s − 6·13-s + 2·17-s + 19-s − 4·23-s − 25-s + 6·29-s − 8·31-s + 2·35-s − 6·37-s − 2·41-s + 2·43-s + 6·45-s − 47-s + 49-s + 6·53-s − 4·55-s + 8·61-s + 3·63-s + 12·65-s − 8·71-s + 6·73-s − 2·77-s + 4·79-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s − 9-s + 0.603·11-s − 1.66·13-s + 0.485·17-s + 0.229·19-s − 0.834·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.338·35-s − 0.986·37-s − 0.312·41-s + 0.304·43-s + 0.894·45-s − 0.145·47-s + 1/7·49-s + 0.824·53-s − 0.539·55-s + 1.02·61-s + 0.377·63-s + 1.48·65-s − 0.949·71-s + 0.702·73-s − 0.227·77-s + 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100016\)    =    \(2^{4} \cdot 7 \cdot 19 \cdot 47\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100016} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 100016,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;19,\;47\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;19,\;47\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
19 \( 1 - T \)
47 \( 1 + T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−13.98277708789507, −13.73754934656253, −12.85273678005658, −12.31013666945780, −12.07415458044576, −11.70347850359433, −11.19950061104166, −10.54808940590206, −9.974399714831191, −9.653730955205994, −8.929173111234216, −8.595891034135915, −7.899781018032967, −7.539588924898920, −7.027816111078743, −6.470963480358664, −5.824264579128496, −5.231528106153468, −4.853238780039189, −3.873022900683446, −3.763348424422795, −2.926041453721965, −2.442655563172087, −1.677807359917766, −0.5954144479175503, 0, 0.5954144479175503, 1.677807359917766, 2.442655563172087, 2.926041453721965, 3.763348424422795, 3.873022900683446, 4.853238780039189, 5.231528106153468, 5.824264579128496, 6.470963480358664, 7.027816111078743, 7.539588924898920, 7.899781018032967, 8.595891034135915, 8.929173111234216, 9.653730955205994, 9.974399714831191, 10.54808940590206, 11.19950061104166, 11.70347850359433, 12.07415458044576, 12.31013666945780, 12.85273678005658, 13.73754934656253, 13.98277708789507

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