Properties

Degree $2$
Conductor $41070$
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 − 8-s + 9-s + 10-s + 4·11-s − 12-s + 2·13-s + 15-s + 16-s − 2·17-s − 18-s − 4·19-s − 20-s − 4·22-s + 8·23-s + 24-s + 25-s − 2·26-s − 27-s + 2·29-s − 30-s − 8·31-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 − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.554·13-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.371·29-s − 0.182·30-s − 1.43·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(41070\)    =    \(2 \cdot 3 \cdot 5 \cdot 37^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{41070} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 41070,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
37 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.17624394082624, −14.63248557339359, −14.12948543470641, −13.24591118664109, −12.82777095634538, −12.40206010915689, −11.62395094340519, −11.26692213660119, −10.98839746155703, −10.38822111910646, −9.661869918342192, −9.171930863289349, −8.634159568958242, −8.305652824773085, −7.307157251509960, −7.034811132167405, −6.434899051590034, −6.000735786719525, −5.151123139470064, −4.545248824164590, −3.839723163269884, −3.328564766759941, −2.362422280347576, −1.534332434323971, −0.9279133791819109, 0, 0.9279133791819109, 1.534332434323971, 2.362422280347576, 3.328564766759941, 3.839723163269884, 4.545248824164590, 5.151123139470064, 6.000735786719525, 6.434899051590034, 7.034811132167405, 7.307157251509960, 8.305652824773085, 8.634159568958242, 9.171930863289349, 9.661869918342192, 10.38822111910646, 10.98839746155703, 11.26692213660119, 11.62395094340519, 12.40206010915689, 12.82777095634538, 13.24591118664109, 14.12948543470641, 14.63248557339359, 15.17624394082624

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