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Properties

Label 2-2900-2900.1527-c0-0-0
Degree $2$
Conductor $2900$
Sign $0.685 + 0.727i$
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  + (−0.393 − 0.919i)2-s + (−0.691 + 0.722i)4-s + (−0.834 − 0.550i)5-s + (0.936 + 0.351i)8-s + (0.963 − 0.266i)9-s + (−0.178 + 0.983i)10-s + (−0.0302 + 0.0331i)13-s + (−0.0448 − 0.998i)16-s + (0.568 + 1.74i)17-s + (−0.623 − 0.781i)18-s + (0.974 − 0.222i)20-s + (0.393 + 0.919i)25-s + (0.0423 + 0.0148i)26-s + (−0.266 + 0.963i)29-s + (−0.900 + 0.433i)32-s + ⋯
L(s)  = 1  + (−0.393 − 0.919i)2-s + (−0.691 + 0.722i)4-s + (−0.834 − 0.550i)5-s + (0.936 + 0.351i)8-s + (0.963 − 0.266i)9-s + (−0.178 + 0.983i)10-s + (−0.0302 + 0.0331i)13-s + (−0.0448 − 0.998i)16-s + (0.568 + 1.74i)17-s + (−0.623 − 0.781i)18-s + (0.974 − 0.222i)20-s + (0.393 + 0.919i)25-s + (0.0423 + 0.0148i)26-s + (−0.266 + 0.963i)29-s + (−0.900 + 0.433i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $0.685 + 0.727i$
Analytic conductor: \(1.44728\)
Root analytic conductor: \(1.20303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (1527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2900,\ (\ :0),\ 0.685 + 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8633620149\)
\(L(\frac12)\) \(\approx\) \(0.8633620149\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.393 + 0.919i)T \)
5 \( 1 + (0.834 + 0.550i)T \)
29 \( 1 + (0.266 - 0.963i)T \)
good3 \( 1 + (-0.963 + 0.266i)T^{2} \)
7 \( 1 + (-0.781 - 0.623i)T^{2} \)
11 \( 1 + (-0.512 + 0.858i)T^{2} \)
13 \( 1 + (0.0302 - 0.0331i)T + (-0.0896 - 0.995i)T^{2} \)
17 \( 1 + (-0.568 - 1.74i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.266 + 0.963i)T^{2} \)
23 \( 1 + (-0.990 + 0.134i)T^{2} \)
31 \( 1 + (0.178 + 0.983i)T^{2} \)
37 \( 1 + (-0.252 - 0.913i)T + (-0.858 + 0.512i)T^{2} \)
41 \( 1 + (0.248 + 1.57i)T + (-0.951 + 0.309i)T^{2} \)
43 \( 1 + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.753 - 0.657i)T^{2} \)
53 \( 1 + (-1.48 + 0.375i)T + (0.880 - 0.473i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.368 + 1.22i)T + (-0.834 + 0.550i)T^{2} \)
67 \( 1 + (-0.657 + 0.753i)T^{2} \)
71 \( 1 + (0.753 - 0.657i)T^{2} \)
73 \( 1 + (-1.50 - 1.31i)T + (0.134 + 0.990i)T^{2} \)
79 \( 1 + (0.919 + 0.393i)T^{2} \)
83 \( 1 + (0.266 - 0.963i)T^{2} \)
89 \( 1 + (-0.502 + 1.25i)T + (-0.722 - 0.691i)T^{2} \)
97 \( 1 + (-0.258 - 0.246i)T + (0.0448 + 0.998i)T^{2} \)
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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

−8.814070963726214525674650845893, −8.304282106863965238823526066058, −7.55944534997364089199555410040, −6.86415987487965082574539336295, −5.58228247559046701886061144139, −4.64611655295412746098952225266, −3.89277888266894446950497601263, −3.42002502481284091281876064037, −1.94132724088739043718925124987, −1.04266319584937700223472406731, 0.846796692168921295817681679413, 2.43582756040273637252298037467, 3.69911509002256548188605803212, 4.48743873136444428471566425828, 5.20414467282569978928227874343, 6.20965818227084589601977356279, 7.05971378833505698993876202916, 7.47353782489031051779050594961, 8.000867651415405796950231529126, 8.975203965107686536814294825759

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