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 + 4·11-s + 12-s − 6·13-s + 14-s + 15-s − 16-s − 2·17-s − 18-s + 20-s + 21-s − 4·22-s − 23-s − 3·24-s + 25-s + 6·26-s − 27-s + 28-s + 6·29-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 + 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 1.66·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.223·20-s + 0.218·21-s − 0.852·22-s − 0.208·23-s − 0.612·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
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 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−19.32814822494133, −18.79261437446573, −17.86491739921481, −17.50339356132564, −16.91712437487854, −16.42598538306978, −15.69610479556662, −14.81011401703528, −14.25499504638350, −13.52451711574221, −12.52811422818326, −12.23673001365380, −11.39675597336763, −10.65612815455132, −9.844520252090504, −9.443263432517823, −8.689826211509155, −7.817613173787654, −7.112216412511400, −6.467573856862762, −5.340675143239995, −4.492591038237952, −3.960586781771923, −2.528796618400089, −1.112908596368453, 0, 1.112908596368453, 2.528796618400089, 3.960586781771923, 4.492591038237952, 5.340675143239995, 6.467573856862762, 7.112216412511400, 7.817613173787654, 8.689826211509155, 9.443263432517823, 9.844520252090504, 10.65612815455132, 11.39675597336763, 12.23673001365380, 12.52811422818326, 13.52451711574221, 14.25499504638350, 14.81011401703528, 15.69610479556662, 16.42598538306978, 16.91712437487854, 17.50339356132564, 17.86491739921481, 18.79261437446573, 19.32814822494133

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