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 + 3-s + 4-s + 5-s + 6-s + 4·7-s + 8-s − 2·9-s + 10-s + 12-s − 2·13-s + 4·14-s + 15-s + 16-s − 3·17-s − 2·18-s − 3·19-s + 20-s + 4·21-s − 3·23-s + 24-s + 25-s − 2·26-s − 5·27-s + 4·28-s − 9·29-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 + 1.51·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.288·12-s − 0.554·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 0.688·19-s + 0.223·20-s + 0.872·21-s − 0.625·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.962·27-s + 0.755·28-s − 1.67·29-s + 0.182·30-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  =  $(2,\ 100010,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.559334168$
$L(\frac12)$  $\approx$  $5.559334168$
$L(\frac{3}{2})$   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 \( 1 - T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−14.00966325691108, −13.24244804090277, −12.95742962646782, −12.40392543369974, −11.64392593339440, −11.37584133918153, −11.01610811404888, −10.46412288576436, −9.735488228594704, −9.285362451107072, −8.658154590387138, −8.262555959710870, −7.727961415262172, −7.337407768999270, −6.594938134028162, −5.986089876448975, −5.474377708532811, −5.091576473105662, −4.296614080483345, −4.093701941294723, −3.241404828864255, −2.451208992674366, −2.128746130247996, −1.689988740693980, −0.5830599871893472, 0.5830599871893472, 1.689988740693980, 2.128746130247996, 2.451208992674366, 3.241404828864255, 4.093701941294723, 4.296614080483345, 5.091576473105662, 5.474377708532811, 5.986089876448975, 6.594938134028162, 7.337407768999270, 7.727961415262172, 8.262555959710870, 8.658154590387138, 9.285362451107072, 9.735488228594704, 10.46412288576436, 11.01610811404888, 11.37584133918153, 11.64392593339440, 12.40392543369974, 12.95742962646782, 13.24244804090277, 14.00966325691108

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