Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23^{2} $
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 + 7-s + 8-s + 9-s − 10-s + 4·11-s + 12-s − 2·13-s + 14-s − 15-s + 16-s − 2·17-s + 18-s − 4·19-s − 20-s + 21-s + 4·22-s + 24-s + 25-s − 2·26-s + 27-s + 28-s − 2·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.218·21-s + 0.852·22-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(111090\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 23^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{111090} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 111090,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.96546928438187, −13.43308012463511, −12.99455584705609, −12.38258156258373, −12.03282273655306, −11.60138596511064, −11.08010447869258, −10.52477898650970, −10.08682257561701, −9.328954420757641, −8.944098948693347, −8.427270919976463, −7.957158722914284, −7.288714819344041, −6.815816773440610, −6.548114685917530, −5.703044239443452, −5.170777988388811, −4.529050301383056, −4.019680429347538, −3.757751591470317, −2.977820592628674, −2.314774338756836, −1.803879633211711, −1.081227839045979, 0, 1.081227839045979, 1.803879633211711, 2.314774338756836, 2.977820592628674, 3.757751591470317, 4.019680429347538, 4.529050301383056, 5.170777988388811, 5.703044239443452, 6.548114685917530, 6.815816773440610, 7.288714819344041, 7.957158722914284, 8.427270919976463, 8.944098948693347, 9.328954420757641, 10.08682257561701, 10.52477898650970, 11.08010447869258, 11.60138596511064, 12.03282273655306, 12.38258156258373, 12.99455584705609, 13.43308012463511, 13.96546928438187

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