Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 23 \cdot 29 $
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 + 5-s + 6-s − 4·7-s + 3·8-s + 9-s − 10-s + 2·11-s + 12-s + 2·13-s + 4·14-s − 15-s − 16-s + 4·17-s − 18-s − 6·19-s − 20-s + 4·21-s − 2·22-s + 23-s − 3·24-s + 25-s − 2·26-s − 27-s + 4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s + 0.554·13-s + 1.06·14-s − 0.258·15-s − 1/4·16-s + 0.970·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s + 0.872·21-s − 0.426·22-s + 0.208·23-s − 0.612·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.755·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10005\)    =    \(3 \cdot 5 \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{10005} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 10005,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;23,\;29\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;23,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−16.94861543965889, −16.56403701970704, −16.05042172763250, −15.44120725496727, −14.34678087974847, −14.26727779940374, −13.18362244214218, −12.80525249020138, −12.65039842671847, −11.53053382801901, −10.96155026166579, −10.21377830294429, −9.907969832216711, −9.309717013313750, −8.804044977615559, −8.118966531423164, −7.191277386292722, −6.681676482682231, −5.971803804521422, −5.521243359431466, −4.450437366888939, −3.859739728006566, −3.081249179015160, −1.861060043820246, −0.9417941610773193, 0, 0.9417941610773193, 1.861060043820246, 3.081249179015160, 3.859739728006566, 4.450437366888939, 5.521243359431466, 5.971803804521422, 6.681676482682231, 7.191277386292722, 8.118966531423164, 8.804044977615559, 9.309717013313750, 9.907969832216711, 10.21377830294429, 10.96155026166579, 11.53053382801901, 12.65039842671847, 12.80525249020138, 13.18362244214218, 14.26727779940374, 14.34678087974847, 15.44120725496727, 16.05042172763250, 16.56403701970704, 16.94861543965889

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