Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 73 \cdot 137 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 4-s + 5-s + 2·6-s − 4·7-s + 8-s + 9-s + 10-s + 2·11-s + 2·12-s + 2·13-s − 4·14-s + 2·15-s + 16-s + 2·17-s + 18-s + 4·19-s + 20-s − 8·21-s + 2·22-s − 4·23-s + 2·24-s + 25-s + 2·26-s − 4·27-s − 4·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s + 0.577·12-s + 0.554·13-s − 1.06·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 1.74·21-s + 0.426·22-s − 0.834·23-s + 0.408·24-s + 1/5·25-s + 0.392·26-s − 0.769·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100010\)    =    \(2 \cdot 5 \cdot 73 \cdot 137\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{100010} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 100010,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.117749770$
$L(\frac12)$  $\approx$  $7.117749770$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;73,\;137\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;73,\;137\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
5 \( 1 - T \)
73 \( 1 + T \)
137 \( 1 - T \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.85282592859169, −13.35612205528751, −12.91729364152457, −12.47765213147414, −11.98752240145763, −11.39185343026111, −10.84070573562196, −10.04963189958537, −9.795241183554139, −9.309576093978146, −8.949589423919180, −8.214067948631447, −7.716652793500672, −7.241661026950722, −6.477704173319681, −6.054357320177607, −5.886456439561804, −4.938120686743562, −4.237564015115748, −3.678860108776172, −3.274054357151089, −2.764702243195555, −2.322131851465763, −1.416548322569268, −0.6945487919878361, 0.6945487919878361, 1.416548322569268, 2.322131851465763, 2.764702243195555, 3.274054357151089, 3.678860108776172, 4.237564015115748, 4.938120686743562, 5.886456439561804, 6.054357320177607, 6.477704173319681, 7.241661026950722, 7.716652793500672, 8.214067948631447, 8.949589423919180, 9.309576093978146, 9.795241183554139, 10.04963189958537, 10.84070573562196, 11.39185343026111, 11.98752240145763, 12.47765213147414, 12.91729364152457, 13.35612205528751, 13.85282592859169

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