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Properties

Label 2-100002-1.1-c1-0-0
Degree $2$
Conductor $100002$
Sign $-1$
Analytic cond. $798.519$
Root an. cond. $28.2580$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 4·5-s + 6-s − 7-s − 8-s + 9-s − 4·10-s − 6·11-s − 12-s + 4·13-s + 14-s − 4·15-s + 16-s − 4·17-s − 18-s + 4·19-s + 4·20-s + 21-s + 6·22-s − 6·23-s + 24-s + 11·25-s − 4·26-s − 27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.26·10-s − 1.80·11-s − 0.288·12-s + 1.10·13-s + 0.267·14-s − 1.03·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.917·19-s + 0.894·20-s + 0.218·21-s + 1.27·22-s − 1.25·23-s + 0.204·24-s + 11/5·25-s − 0.784·26-s − 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100002\)    =    \(2 \cdot 3 \cdot 7 \cdot 2381\)
Sign: $-1$
Analytic conductor: \(798.519\)
Root analytic conductor: \(28.2580\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 100002,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 + T \)
2381 \( 1 + T \)
good5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.75872524338715, −13.47142640840436, −12.98089281832174, −12.79889092955357, −11.97540891545607, −11.35568005906787, −10.82476736529513, −10.48033759776295, −10.13965131032091, −9.615079073394060, −9.153232335842162, −8.677103191653522, −8.053827792529768, −7.419765217235899, −6.937382861423259, −6.235679253533884, −5.970106805573196, −5.342360225710985, −5.217654249064748, −4.178982771470217, −3.390836180293170, −2.692543617042017, −2.087506737518769, −1.721587913068685, −0.8158637626198097, 0, 0.8158637626198097, 1.721587913068685, 2.087506737518769, 2.692543617042017, 3.390836180293170, 4.178982771470217, 5.217654249064748, 5.342360225710985, 5.970106805573196, 6.235679253533884, 6.937382861423259, 7.419765217235899, 8.053827792529768, 8.677103191653522, 9.153232335842162, 9.615079073394060, 10.13965131032091, 10.48033759776295, 10.82476736529513, 11.35568005906787, 11.97540891545607, 12.79889092955357, 12.98089281832174, 13.47142640840436, 13.75872524338715

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