Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 31^{2} $
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 + 7-s − 8-s + 9-s − 10-s − 12-s − 2·13-s − 14-s − 15-s + 16-s + 6·17-s − 18-s + 8·19-s + 20-s − 21-s + 24-s + 25-s + 2·26-s − 27-s + 28-s − 6·29-s + 30-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.218·21-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s + 0.182·30-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(201810\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 31^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{201810} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 201810,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.178756575\)
\(L(\frac12)\)  \(\approx\)  \(2.178756575\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

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

−12.77205654295767, −12.64027698210682, −11.90374350125738, −11.57789209234760, −11.29775968011121, −10.63285071629412, −10.09955168391023, −9.798556706783804, −9.426271783777371, −8.959479035434190, −8.155927298877175, −7.785756289303817, −7.331487634921283, −7.042031457785299, −6.169151528110687, −5.843729018573427, −5.245612902243114, −5.056838816017966, −4.183679447188915, −3.506915077612719, −2.984157093735952, −2.317574245911439, −1.627799195554985, −1.061230659543612, −0.5651036197773037, 0.5651036197773037, 1.061230659543612, 1.627799195554985, 2.317574245911439, 2.984157093735952, 3.506915077612719, 4.183679447188915, 5.056838816017966, 5.245612902243114, 5.843729018573427, 6.169151528110687, 7.042031457785299, 7.331487634921283, 7.785756289303817, 8.155927298877175, 8.959479035434190, 9.426271783777371, 9.798556706783804, 10.09955168391023, 10.63285071629412, 11.29775968011121, 11.57789209234760, 11.90374350125738, 12.64027698210682, 12.77205654295767

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