Properties

Degree $2$
Conductor $100011$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s + 2·7-s + 9-s − 2·10-s + 3·11-s − 2·12-s + 3·13-s + 4·14-s + 15-s − 4·16-s + 17-s + 2·18-s + 19-s − 2·20-s − 2·21-s + 6·22-s − 7·23-s − 4·25-s + 6·26-s − 27-s + 4·28-s − 2·29-s + 2·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 0.755·7-s + 1/3·9-s − 0.632·10-s + 0.904·11-s − 0.577·12-s + 0.832·13-s + 1.06·14-s + 0.258·15-s − 16-s + 0.242·17-s + 0.471·18-s + 0.229·19-s − 0.447·20-s − 0.436·21-s + 1.27·22-s − 1.45·23-s − 4/5·25-s + 1.17·26-s − 0.192·27-s + 0.755·28-s − 0.371·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100011\)    =    \(3 \cdot 17 \cdot 37 \cdot 53\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{100011} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 100011,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
17 \( 1 - T \)
37 \( 1 - T \)
53 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
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

−14.00142162515245, −13.66962678089508, −13.00465651726595, −12.49624506959137, −11.90725325814974, −11.79988443851420, −11.36649457301913, −10.81195006445202, −10.30138748714382, −9.524832499037356, −9.099955791928274, −8.422627715996364, −7.785189938367988, −7.501535642468895, −6.499642309846511, −6.323211604990791, −5.839821333318269, −5.167089079160946, −4.785156868729121, −4.041826577958456, −3.848005368634488, −3.335403628845769, −2.335363561446685, −1.756924577160709, −1.008374391004073, 0, 1.008374391004073, 1.756924577160709, 2.335363561446685, 3.335403628845769, 3.848005368634488, 4.041826577958456, 4.785156868729121, 5.167089079160946, 5.839821333318269, 6.323211604990791, 6.499642309846511, 7.501535642468895, 7.785189938367988, 8.422627715996364, 9.099955791928274, 9.524832499037356, 10.30138748714382, 10.81195006445202, 11.36649457301913, 11.79988443851420, 11.90725325814974, 12.49624506959137, 13.00465651726595, 13.66962678089508, 14.00142162515245

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