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  + 2.92i·3-s − 2.77i·7-s − 5.58·9-s + 2.02·11-s − 3.10i·13-s + 7.91i·17-s − 2.17·19-s + 8.13·21-s − 4.75i·23-s − 7.56i·27-s − 3.85·29-s − 10.6·31-s + 5.94i·33-s − 6.58i·37-s + 9.10·39-s + ⋯
L(s)  = 1  + 1.69i·3-s − 1.05i·7-s − 1.86·9-s + 0.611·11-s − 0.862i·13-s + 1.91i·17-s − 0.500·19-s + 1.77·21-s − 0.990i·23-s − 1.45i·27-s − 0.716·29-s − 1.91·31-s + 1.03i·33-s − 1.08i·37-s + 1.45·39-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

\(L(1)\) \(\approx\) \(0.6917878118\)
\(L(\frac12)\) \(\approx\) \(0.6917878118\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.92iT - 3T^{2} \)
7 \( 1 + 2.77iT - 7T^{2} \)
11 \( 1 - 2.02T + 11T^{2} \)
13 \( 1 + 3.10iT - 13T^{2} \)
17 \( 1 - 7.91iT - 17T^{2} \)
19 \( 1 + 2.17T + 19T^{2} \)
23 \( 1 + 4.75iT - 23T^{2} \)
29 \( 1 + 3.85T + 29T^{2} \)
31 \( 1 + 10.6T + 31T^{2} \)
37 \( 1 + 6.58iT - 37T^{2} \)
43 \( 1 - 4.80iT - 43T^{2} \)
47 \( 1 + 4.03iT - 47T^{2} \)
53 \( 1 + 1.94iT - 53T^{2} \)
59 \( 1 + 4.75T + 59T^{2} \)
61 \( 1 - 2.89T + 61T^{2} \)
67 \( 1 + 5.97iT - 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 + 9.24iT - 73T^{2} \)
79 \( 1 - 0.320T + 79T^{2} \)
83 \( 1 + 3.69iT - 83T^{2} \)
89 \( 1 + 8.96T + 89T^{2} \)
97 \( 1 + 17.8iT - 97T^{2} \)
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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

−8.454092137629094469067286496501, −7.75081789415349957915712134815, −6.73497710803627552135169620892, −5.88895571405034976720914001389, −5.23626422440470625301443069558, −4.13012131570272207856858916725, −3.99012167463181810971845859823, −3.22703048381050470202608454077, −1.81280879084086269723504677444, −0.19601820553194928549736211281, 1.23257228995306367619353025466, 2.06970770955272806239507509046, 2.72141268950530841983903766108, 3.83119443908376276242777235751, 5.17249040278324061113000847788, 5.67071194079131129822810178427, 6.61325445327831862010530796247, 7.01540097359830275033172712583, 7.66437237206778763037580563635, 8.482451685446989399355469381285

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