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  − 2.21i·3-s − 5.06i·7-s − 1.90·9-s − 2.55·11-s + 4.93i·13-s + 2.68i·17-s − 4.72·19-s − 11.2·21-s + 1.49i·23-s − 2.41i·27-s − 2.43·29-s + 3.19·31-s + 5.67i·33-s + 2.90i·37-s + 10.9·39-s + ⋯
L(s)  = 1  − 1.27i·3-s − 1.91i·7-s − 0.636·9-s − 0.771·11-s + 1.36i·13-s + 0.651i·17-s − 1.08·19-s − 2.44·21-s + 0.311i·23-s − 0.465i·27-s − 0.451·29-s + 0.573·31-s + 0.987i·33-s + 0.478i·37-s + 1.75·39-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

\(L(1)\) \(\approx\) \(0.2805154092\)
\(L(\frac12)\) \(\approx\) \(0.2805154092\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.21iT - 3T^{2} \)
7 \( 1 + 5.06iT - 7T^{2} \)
11 \( 1 + 2.55T + 11T^{2} \)
13 \( 1 - 4.93iT - 13T^{2} \)
17 \( 1 - 2.68iT - 17T^{2} \)
19 \( 1 + 4.72T + 19T^{2} \)
23 \( 1 - 1.49iT - 23T^{2} \)
29 \( 1 + 2.43T + 29T^{2} \)
31 \( 1 - 3.19T + 31T^{2} \)
37 \( 1 - 2.90iT - 37T^{2} \)
43 \( 1 - 7.62iT - 43T^{2} \)
47 \( 1 - 5.15iT - 47T^{2} \)
53 \( 1 - 11.1iT - 53T^{2} \)
59 \( 1 + 1.49T + 59T^{2} \)
61 \( 1 - 1.06T + 61T^{2} \)
67 \( 1 - 10.5iT - 67T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 + 8.28iT - 73T^{2} \)
79 \( 1 - 4.28T + 79T^{2} \)
83 \( 1 + 10.5iT - 83T^{2} \)
89 \( 1 + 9.36T + 89T^{2} \)
97 \( 1 + 3.37iT - 97T^{2} \)
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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

−8.185522916081055541125251910652, −7.73841724511835007547761677733, −7.07825926948405559634864023052, −6.60340729685096424960021235362, −5.93183669271804193892566885154, −4.44885601411351677678900631740, −4.24901197689745335932155434781, −2.98717680689894999762254306896, −1.82520777247152123741553858143, −1.19226306949221549833061345847, 0.079723207578526011100499935300, 2.16691143668347396865640440920, 2.82043650841908994701810052938, 3.61991917364469964722336342811, 4.72171079604294707217546263522, 5.34551333896264484511280943542, 5.67630851728877711506390553157, 6.68833894505254560158593753280, 7.85451807921042455915256289369, 8.524121729601726445899243735340

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