Properties

Degree 2
Conductor $ 2 \cdot 5^{2} \cdot 7 \cdot 293 $
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 + 6-s − 7-s + 8-s − 2·9-s − 6·11-s + 12-s + 3·13-s − 14-s + 16-s + 4·17-s − 2·18-s − 8·19-s − 21-s − 6·22-s + 6·23-s + 24-s + 3·26-s − 5·27-s − 28-s − 10·31-s + 32-s − 6·33-s + 4·34-s − 2·36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 1.80·11-s + 0.288·12-s + 0.832·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.471·18-s − 1.83·19-s − 0.218·21-s − 1.27·22-s + 1.25·23-s + 0.204·24-s + 0.588·26-s − 0.962·27-s − 0.188·28-s − 1.79·31-s + 0.176·32-s − 1.04·33-s + 0.685·34-s − 1/3·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(102550\)    =    \(2 \cdot 5^{2} \cdot 7 \cdot 293\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{102550} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 102550,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7,\;293\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7,\;293\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
293 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−14.01579846484678, −13.38457695473415, −12.99889134187530, −12.66228774026590, −12.39419265148265, −11.33100291547162, −11.07284699910772, −10.64994137547010, −10.24252038009459, −9.446909184297066, −8.968256669638632, −8.379115490130526, −8.082706942459123, −7.418170260694881, −6.989666679020359, −6.254058617051143, −5.637819754245285, −5.453975401263257, −4.779949839642795, −3.990056380583454, −3.467174794827074, −3.068571717710002, −2.304702705277335, −2.086925994654535, −0.8819139902230116, 0, 0.8819139902230116, 2.086925994654535, 2.304702705277335, 3.068571717710002, 3.467174794827074, 3.990056380583454, 4.779949839642795, 5.453975401263257, 5.637819754245285, 6.254058617051143, 6.989666679020359, 7.418170260694881, 8.082706942459123, 8.379115490130526, 8.968256669638632, 9.446909184297066, 10.24252038009459, 10.64994137547010, 11.07284699910772, 11.33100291547162, 12.39419265148265, 12.66228774026590, 12.99889134187530, 13.38457695473415, 14.01579846484678

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