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Properties

Label 2-2900-5.4-c1-0-41
Degree $2$
Conductor $2900$
Sign $-0.447 - 0.894i$
Analytic cond. $23.1566$
Root an. cond. $4.81213$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s − 4i·7-s − 6·9-s − 11-s − 3i·13-s − 2i·17-s − 4·19-s − 12·21-s − 6i·23-s + 9i·27-s + 29-s + 9·31-s + 3i·33-s + 8i·37-s − 9·39-s + ⋯
L(s)  = 1  − 1.73i·3-s − 1.51i·7-s − 2·9-s − 0.301·11-s − 0.832i·13-s − 0.485i·17-s − 0.917·19-s − 2.61·21-s − 1.25i·23-s + 1.73i·27-s + 0.185·29-s + 1.61·31-s + 0.522i·33-s + 1.31i·37-s − 1.44·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(23.1566\)
Root analytic conductor: \(4.81213\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2900,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.147352786\)
\(L(\frac12)\) \(\approx\) \(1.147352786\)
\(L(\frac{3}{2})\) 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 \( 1 \)
29 \( 1 - T \)
good3 \( 1 + 3iT - 3T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + 3iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
31 \( 1 - 9T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + 5iT - 43T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 + 5iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.255516315186511054408977969745, −7.44413148945122034190638266403, −6.68259705694532716113960154904, −6.46612785139959394339708704386, −5.24838490168089429545923566185, −4.33516336110932363381446092258, −3.14188537352762663975550552415, −2.30850990335944994427550041726, −1.10074789646366179681599013388, −0.39662715908482398433421750941, 2.05270465277567400135754719590, 2.91920140040291001908349715597, 3.86813069012593587202929275985, 4.57021771834001088748463699249, 5.41223417929931121759785931222, 5.86546189692156582120155029805, 6.84776543516457636317410813253, 8.281441868506121524829947280501, 8.624867548541186202168076388516, 9.318811459050558113699128859529

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