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Properties

Label 2-2900-2900.1083-c0-0-0
Degree $2$
Conductor $2900$
Sign $-0.884 - 0.466i$
Analytic cond. $1.44728$
Root an. cond. $1.20303$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.995 + 0.0896i)2-s + (0.983 − 0.178i)4-s + (−0.998 + 0.0448i)5-s + (−0.963 + 0.266i)8-s + (0.550 + 0.834i)9-s + (0.990 − 0.134i)10-s + (−0.679 + 1.49i)13-s + (0.936 − 0.351i)16-s + (−0.145 − 0.105i)17-s + (−0.623 − 0.781i)18-s + (−0.974 + 0.222i)20-s + (0.995 − 0.0896i)25-s + (0.542 − 1.55i)26-s + (−0.834 − 0.550i)29-s + (−0.900 + 0.433i)32-s + ⋯
L(s)  = 1  + (−0.995 + 0.0896i)2-s + (0.983 − 0.178i)4-s + (−0.998 + 0.0448i)5-s + (−0.963 + 0.266i)8-s + (0.550 + 0.834i)9-s + (0.990 − 0.134i)10-s + (−0.679 + 1.49i)13-s + (0.936 − 0.351i)16-s + (−0.145 − 0.105i)17-s + (−0.623 − 0.781i)18-s + (−0.974 + 0.222i)20-s + (0.995 − 0.0896i)25-s + (0.542 − 1.55i)26-s + (−0.834 − 0.550i)29-s + (−0.900 + 0.433i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $-0.884 - 0.466i$
Analytic conductor: \(1.44728\)
Root analytic conductor: \(1.20303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (1083, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2900,\ (\ :0),\ -0.884 - 0.466i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2986212941\)
\(L(\frac12)\) \(\approx\) \(0.2986212941\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.995 - 0.0896i)T \)
5 \( 1 + (0.998 - 0.0448i)T \)
29 \( 1 + (0.834 + 0.550i)T \)
good3 \( 1 + (-0.550 - 0.834i)T^{2} \)
7 \( 1 + (0.781 + 0.623i)T^{2} \)
11 \( 1 + (-0.919 + 0.393i)T^{2} \)
13 \( 1 + (0.679 - 1.49i)T + (-0.657 - 0.753i)T^{2} \)
17 \( 1 + (0.145 + 0.105i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.834 - 0.550i)T^{2} \)
23 \( 1 + (-0.880 - 0.473i)T^{2} \)
31 \( 1 + (-0.990 - 0.134i)T^{2} \)
37 \( 1 + (1.15 - 0.761i)T + (0.393 - 0.919i)T^{2} \)
41 \( 1 + (0.869 + 1.70i)T + (-0.587 + 0.809i)T^{2} \)
43 \( 1 + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.858 + 0.512i)T^{2} \)
53 \( 1 + (0.0500 - 0.124i)T + (-0.722 - 0.691i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.0447 - 1.99i)T + (-0.998 - 0.0448i)T^{2} \)
67 \( 1 + (-0.512 - 0.858i)T^{2} \)
71 \( 1 + (0.858 + 0.512i)T^{2} \)
73 \( 1 + (1.29 - 0.772i)T + (0.473 - 0.880i)T^{2} \)
79 \( 1 + (0.0896 - 0.995i)T^{2} \)
83 \( 1 + (0.834 + 0.550i)T^{2} \)
89 \( 1 + (0.530 - 0.634i)T + (-0.178 - 0.983i)T^{2} \)
97 \( 1 + (0.353 + 1.95i)T + (-0.936 + 0.351i)T^{2} \)
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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

−9.091355234971263335982635285645, −8.563109348676676141734649232725, −7.72706783339400233488451826654, −7.12257924951323865057856590701, −6.74288291603977695066736545244, −5.45446698868516882245364538832, −4.56194801875138933754652355890, −3.70132432101656094412719127618, −2.46302474202346325307217668479, −1.59741330090860427201199099664, 0.25936133602006171255763402372, 1.55570684662329412678327269503, 3.03178555531540538871816608262, 3.50468387455152395918491817667, 4.68278097287946704274327307699, 5.70637856411125368382468329766, 6.70874562201545407957350887423, 7.28638081236237472585752920817, 7.958800357022271611476117909309, 8.521987347751198176691188205249

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