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Properties

Label 2-2900-116.111-c0-0-1
Degree $2$
Conductor $2900$
Sign $-0.833 - 0.552i$
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  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.900 − 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.400 − 0.193i)13-s + (−0.900 − 0.433i)16-s − 1.24·17-s + (0.900 + 0.433i)18-s + (0.0990 + 0.433i)26-s + (−0.900 − 0.433i)29-s + (0.222 + 0.974i)32-s + (0.777 + 0.974i)34-s + (−0.222 − 0.974i)36-s + (−0.400 + 0.193i)37-s − 0.445·41-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.900 − 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.400 − 0.193i)13-s + (−0.900 − 0.433i)16-s − 1.24·17-s + (0.900 + 0.433i)18-s + (0.0990 + 0.433i)26-s + (−0.900 − 0.433i)29-s + (0.222 + 0.974i)32-s + (0.777 + 0.974i)34-s + (−0.222 − 0.974i)36-s + (−0.400 + 0.193i)37-s − 0.445·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $-0.833 - 0.552i$
Analytic conductor: \(1.44728\)
Root analytic conductor: \(1.20303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (1851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2900,\ (\ :0),\ -0.833 - 0.552i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02348676015\)
\(L(\frac12)\) \(\approx\) \(0.02348676015\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 \)
29 \( 1 + (0.900 + 0.433i)T \)
good3 \( 1 + (0.900 - 0.433i)T^{2} \)
7 \( 1 + (0.900 - 0.433i)T^{2} \)
11 \( 1 + (-0.623 - 0.781i)T^{2} \)
13 \( 1 + (0.400 + 0.193i)T + (0.623 + 0.781i)T^{2} \)
17 \( 1 + 1.24T + T^{2} \)
19 \( 1 + (0.900 + 0.433i)T^{2} \)
23 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.222 - 0.974i)T^{2} \)
37 \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \)
41 \( 1 + 0.445T + T^{2} \)
43 \( 1 + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.623 - 0.781i)T^{2} \)
53 \( 1 + (0.777 + 0.974i)T + (-0.222 + 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
67 \( 1 + (-0.623 + 0.781i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (0.777 - 0.974i)T + (-0.222 - 0.974i)T^{2} \)
79 \( 1 + (-0.623 + 0.781i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
97 \( 1 + (0.0990 - 0.433i)T + (-0.900 - 0.433i)T^{2} \)
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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

−8.544610366557019973190812239636, −8.023423285739517806243763089947, −7.19182114163573067410761260317, −6.34306467965210426880863380082, −5.24551742256432338421566859792, −4.47458420918288846892971084519, −3.45405242804643885651139194984, −2.59571792646187032551193750262, −1.78369969996830180303311029391, −0.01696295817122953092172168749, 1.67474582853928150343523413674, 2.79878377654525726548737233491, 4.06380059061222361153968355358, 4.97288237262447056113967230853, 5.73418989224480668312063377797, 6.48259809398434799020635578621, 7.10559987432271735834708850438, 7.927894629879193368430040630452, 8.740279417581628100451943843534, 9.128743463343243409361845446751

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