Properties

Degree $2$
Conductor $287490$
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 − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 12-s + 2·13-s + 14-s + 15-s + 16-s − 2·17-s − 18-s − 20-s + 21-s + 8·23-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 − 0.485·17-s − 0.235·18-s − 0.223·20-s + 0.218·21-s + 1.66·23-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287490\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 37^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{287490} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 287490,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
37 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.86937337392077, −12.38625870482311, −12.01982266552587, −11.33020165295758, −11.15180778087811, −10.69445228336558, −10.36321424851527, −9.644651071498547, −9.209858585674426, −8.908911555187173, −8.407632059151907, −7.759900671203722, −7.339789698882121, −6.889068347030491, −6.568913658105378, −5.795922871976810, −5.586458347892142, −4.874009233442232, −4.305645931496327, −3.699518635248301, −3.310739867240531, −2.484690192524012, −2.087624711720262, −1.068302787059973, −0.8174382477604706, 0, 0.8174382477604706, 1.068302787059973, 2.087624711720262, 2.484690192524012, 3.310739867240531, 3.699518635248301, 4.305645931496327, 4.874009233442232, 5.586458347892142, 5.795922871976810, 6.568913658105378, 6.889068347030491, 7.339789698882121, 7.759900671203722, 8.407632059151907, 8.908911555187173, 9.209858585674426, 9.644651071498547, 10.36321424851527, 10.69445228336558, 11.15180778087811, 11.33020165295758, 12.01982266552587, 12.38625870482311, 12.86937337392077

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