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 + 3·8-s + 9-s − 10-s + 12-s + 6·13-s − 15-s − 16-s + 6·17-s − 18-s − 4·19-s − 20-s − 23-s − 3·24-s + 25-s − 6·26-s − 27-s + 29-s + 30-s + 4·31-s − 5·32-s − 6·34-s − 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 1.66·13-s − 0.258·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.208·23-s − 0.612·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.185·29-s + 0.182·30-s + 0.718·31-s − 0.883·32-s − 1.02·34-s − 1/6·36-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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−16.88644428130964, −16.64227392130719, −15.98454716216685, −15.37719515724761, −14.53424262090718, −14.01054207083566, −13.38015187196821, −13.02620300029173, −12.24992405011988, −11.62422815269201, −10.86988005922789, −10.34971449187829, −10.01515990262597, −9.225012542492432, −8.612667083519166, −8.138961488895020, −7.465487784512962, −6.498317784933029, −6.067242633235919, −5.319905779295503, −4.621152616997111, −3.852862596770265, −3.083669522176013, −1.620927367156824, −1.215602962736103, 0, 1.215602962736103, 1.620927367156824, 3.083669522176013, 3.852862596770265, 4.621152616997111, 5.319905779295503, 6.067242633235919, 6.498317784933029, 7.465487784512962, 8.138961488895020, 8.612667083519166, 9.225012542492432, 10.01515990262597, 10.34971449187829, 10.86988005922789, 11.62422815269201, 12.24992405011988, 13.02620300029173, 13.38015187196821, 14.01054207083566, 14.53424262090718, 15.37719515724761, 15.98454716216685, 16.64227392130719, 16.88644428130964

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