Properties

Degree $2$
Conductor $141610$
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 + 4-s − 5-s + 8-s − 3·9-s − 10-s − 4·11-s + 6·13-s + 16-s − 3·18-s − 20-s − 4·22-s + 25-s + 6·26-s − 6·29-s + 8·31-s + 32-s − 3·36-s + 10·37-s − 40-s + 2·41-s + 4·43-s − 4·44-s + 3·45-s − 8·47-s + 50-s + 6·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 9-s − 0.316·10-s − 1.20·11-s + 1.66·13-s + 1/4·16-s − 0.707·18-s − 0.223·20-s − 0.852·22-s + 1/5·25-s + 1.17·26-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 1/2·36-s + 1.64·37-s − 0.158·40-s + 0.312·41-s + 0.609·43-s − 0.603·44-s + 0.447·45-s − 1.16·47-s + 0.141·50-s + 0.832·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 141610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(141610\)    =    \(2 \cdot 5 \cdot 7^{2} \cdot 17^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{141610} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 141610,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
17 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.56906476922445, −13.13801783813382, −12.92068874189312, −12.13125657331026, −11.78325289943517, −11.14198055556632, −10.95513158319005, −10.61166668391830, −9.769291181990474, −9.321528337078299, −8.614179612361521, −8.110038904916629, −7.941466626757171, −7.313959185762944, −6.467735063026448, −6.188831678528398, −5.674502303695478, −5.189211907106571, −4.582652371517277, −3.978342293957579, −3.507987305312335, −2.773622867015106, −2.610576870708447, −1.603629055576995, −0.8670152567736104, 0, 0.8670152567736104, 1.603629055576995, 2.610576870708447, 2.773622867015106, 3.507987305312335, 3.978342293957579, 4.582652371517277, 5.189211907106571, 5.674502303695478, 6.188831678528398, 6.467735063026448, 7.313959185762944, 7.941466626757171, 8.110038904916629, 8.614179612361521, 9.321528337078299, 9.769291181990474, 10.61166668391830, 10.95513158319005, 11.14198055556632, 11.78325289943517, 12.13125657331026, 12.92068874189312, 13.13801783813382, 13.56906476922445

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