Properties

Label 2-100016-1.1-c1-0-2
Degree $2$
Conductor $100016$
Sign $1$
Analytic cond. $798.631$
Root an. cond. $28.2600$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·5-s − 7-s − 2·9-s − 3·11-s − 2·13-s + 3·15-s + 2·17-s + 19-s + 21-s + 4·23-s + 4·25-s + 5·27-s − 2·29-s − 10·31-s + 3·33-s + 3·35-s + 37-s + 2·39-s + 5·41-s + 9·43-s + 6·45-s + 47-s + 49-s − 2·51-s − 53-s + 9·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.554·13-s + 0.774·15-s + 0.485·17-s + 0.229·19-s + 0.218·21-s + 0.834·23-s + 4/5·25-s + 0.962·27-s − 0.371·29-s − 1.79·31-s + 0.522·33-s + 0.507·35-s + 0.164·37-s + 0.320·39-s + 0.780·41-s + 1.37·43-s + 0.894·45-s + 0.145·47-s + 1/7·49-s − 0.280·51-s − 0.137·53-s + 1.21·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100016\)    =    \(2^{4} \cdot 7 \cdot 19 \cdot 47\)
Sign: $1$
Analytic conductor: \(798.631\)
Root analytic conductor: \(28.2600\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 100016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5253777014\)
\(L(\frac12)\) \(\approx\) \(0.5253777014\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
19 \( 1 - T \)
47 \( 1 - T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.68210818371677, −13.13267510907111, −12.64143053493839, −12.16836579910634, −11.95177291001027, −11.18933701770289, −10.83287236432298, −10.69988041080656, −9.763163173831213, −9.265047991666486, −8.837024726296776, −7.974568176282039, −7.862922042998365, −7.191194453886076, −6.884661828592859, −5.972609822312327, −5.518017757218439, −5.161168647164755, −4.418786611394743, −3.911008084057703, −3.184263783586308, −2.849109349942583, −2.050285177248228, −0.9020584898205815, −0.3004301763428371, 0.3004301763428371, 0.9020584898205815, 2.050285177248228, 2.849109349942583, 3.184263783586308, 3.911008084057703, 4.418786611394743, 5.161168647164755, 5.518017757218439, 5.972609822312327, 6.884661828592859, 7.191194453886076, 7.862922042998365, 7.974568176282039, 8.837024726296776, 9.265047991666486, 9.763163173831213, 10.69988041080656, 10.83287236432298, 11.18933701770289, 11.95177291001027, 12.16836579910634, 12.64143053493839, 13.13267510907111, 13.68210818371677

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