Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13^{2} $
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 + 9-s − 12-s + 14-s + 16-s − 6·17-s + 18-s + 4·19-s − 21-s − 24-s − 27-s + 28-s + 6·29-s + 4·31-s + 32-s − 6·34-s + 36-s − 10·37-s + 4·38-s − 6·41-s − 42-s − 8·43-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 + 1/3·9-s − 0.288·12-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.218·21-s − 0.204·24-s − 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 1.64·37-s + 0.648·38-s − 0.937·41-s − 0.154·42-s − 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(177450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{177450} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 177450,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
13 \( 1 \)
good11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.40588133410007, −13.04474930624650, −12.40135219222137, −11.90577616344867, −11.61748084931817, −11.28161342145214, −10.63513174898720, −10.13703313793676, −9.929915291700382, −9.055169646455812, −8.433908076657272, −8.335060091695613, −7.355195760239344, −6.997873759014878, −6.599531935476539, −6.136270013248679, −5.330282314091730, −5.096030181322408, −4.675436381778349, −3.949393319733394, −3.575090464331343, −2.758758130897950, −2.247996697194640, −1.563258665219088, −0.9019550316441548, 0, 0.9019550316441548, 1.563258665219088, 2.247996697194640, 2.758758130897950, 3.575090464331343, 3.949393319733394, 4.675436381778349, 5.096030181322408, 5.330282314091730, 6.136270013248679, 6.599531935476539, 6.997873759014878, 7.355195760239344, 8.335060091695613, 8.433908076657272, 9.055169646455812, 9.929915291700382, 10.13703313793676, 10.63513174898720, 11.28161342145214, 11.61748084931817, 11.90577616344867, 12.40135219222137, 13.04474930624650, 13.40588133410007

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