Properties

Degree $2$
Conductor $4100$
Sign $0.447 + 0.894i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.0950i·3-s − 3.14i·7-s + 2.99·9-s − 1.67·11-s + 6.63i·13-s − 5.16i·17-s + 4.72·19-s + 0.299·21-s − 8.82i·23-s + 0.569i·27-s + 1.80·29-s − 1.65·31-s − 0.159i·33-s + 1.99i·37-s − 0.630·39-s + ⋯
L(s)  = 1  + 0.0549i·3-s − 1.18i·7-s + 0.996·9-s − 0.505·11-s + 1.83i·13-s − 1.25i·17-s + 1.08·19-s + 0.0652·21-s − 1.83i·23-s + 0.109i·27-s + 0.336·29-s − 0.298·31-s − 0.0277i·33-s + 0.327i·37-s − 0.100·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4100\)    =    \(2^{2} \cdot 5^{2} \cdot 41\)
Sign: $0.447 + 0.894i$
Motivic weight: \(1\)
Character: $\chi_{4100} (1149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4100,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.918330543\)
\(L(\frac12)\) \(\approx\) \(1.918330543\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
41 \( 1 + T \)
good3 \( 1 - 0.0950iT - 3T^{2} \)
7 \( 1 + 3.14iT - 7T^{2} \)
11 \( 1 + 1.67T + 11T^{2} \)
13 \( 1 - 6.63iT - 13T^{2} \)
17 \( 1 + 5.16iT - 17T^{2} \)
19 \( 1 - 4.72T + 19T^{2} \)
23 \( 1 + 8.82iT - 23T^{2} \)
29 \( 1 - 1.80T + 29T^{2} \)
31 \( 1 + 1.65T + 31T^{2} \)
37 \( 1 - 1.99iT - 37T^{2} \)
43 \( 1 - 1.46iT - 43T^{2} \)
47 \( 1 - 8.53iT - 47T^{2} \)
53 \( 1 + 9.35iT - 53T^{2} \)
59 \( 1 + 8.82T + 59T^{2} \)
61 \( 1 - 12.6T + 61T^{2} \)
67 \( 1 + 9.67iT - 67T^{2} \)
71 \( 1 + 0.776T + 71T^{2} \)
73 \( 1 - 8.33iT - 73T^{2} \)
79 \( 1 + 0.915T + 79T^{2} \)
83 \( 1 + 10.0iT - 83T^{2} \)
89 \( 1 - 6.44T + 89T^{2} \)
97 \( 1 - 8.32iT - 97T^{2} \)
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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

−8.195670319961422113587398497009, −7.36067209846619659003757840120, −6.95647611652620490649737993810, −6.37920741521496932270477686977, −4.98117617033124737147901052651, −4.55339065830231545331046780650, −3.86772999576708696731385667230, −2.78493096707027308487155970811, −1.68054963858447922746781576200, −0.62605776915274697799387293831, 1.10682322114801809465555024742, 2.14316691640828768757521265591, 3.14390690256427304865427339336, 3.80515991574207519714244434373, 5.13294528643273390318889038436, 5.50127480233327155227430483408, 6.15749101847444332824207168628, 7.35189384823055744855410013296, 7.74727413279815474833980505477, 8.493856915771222576753194924678

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