Properties

Degree 2
Conductor $ 2 \cdot 5^{2} \cdot 7 \cdot 617 $
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 + 3·3-s + 4-s − 3·6-s − 7-s − 8-s + 6·9-s − 2·11-s + 3·12-s + 7·13-s + 14-s + 16-s − 4·17-s − 6·18-s − 19-s − 3·21-s + 2·22-s − 3·23-s − 3·24-s − 7·26-s + 9·27-s − 28-s + 9·29-s + 4·31-s − 32-s − 6·33-s + 4·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s − 0.603·11-s + 0.866·12-s + 1.94·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 1.41·18-s − 0.229·19-s − 0.654·21-s + 0.426·22-s − 0.625·23-s − 0.612·24-s − 1.37·26-s + 1.73·27-s − 0.188·28-s + 1.67·29-s + 0.718·31-s − 0.176·32-s − 1.04·33-s + 0.685·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 215950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(215950\)    =    \(2 \cdot 5^{2} \cdot 7 \cdot 617\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{215950} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 215950,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7,\;617\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7,\;617\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
617 \( 1 + T \)
good3 \( 1 - p T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−13.35824736947571, −12.76704311045960, −12.53786776904590, −11.79692721373548, −11.13184888566251, −10.76457364507186, −10.27025336803553, −9.888351193104595, −9.314633543373411, −8.814744493016050, −8.541038015466835, −8.266081653454596, −7.711751919178622, −7.233920154831199, −6.568927229083230, −6.255060134878773, −5.713138512013731, −4.661514103175198, −4.299253703022445, −3.653822206511109, −3.204116819964089, −2.641500165501429, −2.266441350803315, −1.483830411017685, −1.046653160572820, 0, 1.046653160572820, 1.483830411017685, 2.266441350803315, 2.641500165501429, 3.204116819964089, 3.653822206511109, 4.299253703022445, 4.661514103175198, 5.713138512013731, 6.255060134878773, 6.568927229083230, 7.233920154831199, 7.711751919178622, 8.266081653454596, 8.541038015466835, 8.814744493016050, 9.314633543373411, 9.888351193104595, 10.27025336803553, 10.76457364507186, 11.13184888566251, 11.79692721373548, 12.53786776904590, 12.76704311045960, 13.35824736947571

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