Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 23 $
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 + 3·8-s + 9-s − 10-s − 12-s + 2·13-s + 14-s + 15-s − 16-s − 6·17-s − 18-s − 4·19-s − 20-s − 21-s − 23-s + 3·24-s + 25-s − 2·26-s + 27-s + 28-s − 2·29-s − 30-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 + 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.554·13-s + 0.267·14-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 − 0.218·21-s − 0.208·23-s + 0.612·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{2415} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 2415,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−19.24614783274402, −18.72598571056318, −18.07963284344704, −17.58545343661115, −16.93810113284969, −16.25427444766603, −15.54322630193912, −14.87444410087556, −13.98495475258389, −13.58999673131849, −12.96271140514781, −12.39635885594369, −10.99866934036056, −10.73019169072567, −9.797931284439675, −9.208297701677041, −8.721380936062501, −8.102230233277390, −7.139179206482626, −6.447314573281281, −5.431500803263322, −4.392356292424719, −3.755629006339341, −2.456509364881546, −1.538654652219953, 0, 1.538654652219953, 2.456509364881546, 3.755629006339341, 4.392356292424719, 5.431500803263322, 6.447314573281281, 7.139179206482626, 8.102230233277390, 8.721380936062501, 9.208297701677041, 9.797931284439675, 10.73019169072567, 10.99866934036056, 12.39635885594369, 12.96271140514781, 13.58999673131849, 13.98495475258389, 14.87444410087556, 15.54322630193912, 16.25427444766603, 16.93810113284969, 17.58545343661115, 18.07963284344704, 18.72598571056318, 19.24614783274402

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