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Properties

Label 2-2900-116.103-c0-0-1
Degree $2$
Conductor $2900$
Sign $0.995 - 0.0954i$
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  + (0.433 − 0.900i)2-s + (1.40 + 1.12i)3-s + (−0.623 − 0.781i)4-s + (1.62 − 0.781i)6-s + (0.974 + 0.777i)7-s + (−0.974 + 0.222i)8-s + (0.500 + 2.19i)9-s − 1.80i·12-s + (1.12 − 0.541i)14-s + (−0.222 + 0.974i)16-s + (2.19 + 0.499i)18-s + (0.500 + 2.19i)21-s + (−0.781 − 1.62i)23-s + (−1.62 − 0.781i)24-s + (−0.974 + 2.02i)27-s − 1.24i·28-s + ⋯
L(s)  = 1  + (0.433 − 0.900i)2-s + (1.40 + 1.12i)3-s + (−0.623 − 0.781i)4-s + (1.62 − 0.781i)6-s + (0.974 + 0.777i)7-s + (−0.974 + 0.222i)8-s + (0.500 + 2.19i)9-s − 1.80i·12-s + (1.12 − 0.541i)14-s + (−0.222 + 0.974i)16-s + (2.19 + 0.499i)18-s + (0.500 + 2.19i)21-s + (−0.781 − 1.62i)23-s + (−1.62 − 0.781i)24-s + (−0.974 + 2.02i)27-s − 1.24i·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $0.995 - 0.0954i$
Analytic conductor: \(1.44728\)
Root analytic conductor: \(1.20303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2900,\ (\ :0),\ 0.995 - 0.0954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.449532039\)
\(L(\frac12)\) \(\approx\) \(2.449532039\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.433 + 0.900i)T \)
5 \( 1 \)
29 \( 1 + (0.623 - 0.781i)T \)
good3 \( 1 + (-1.40 - 1.12i)T + (0.222 + 0.974i)T^{2} \)
7 \( 1 + (-0.974 - 0.777i)T + (0.222 + 0.974i)T^{2} \)
11 \( 1 + (0.900 + 0.433i)T^{2} \)
13 \( 1 + (-0.900 - 0.433i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.222 - 0.974i)T^{2} \)
23 \( 1 + (0.781 + 1.62i)T + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.623 - 0.781i)T^{2} \)
37 \( 1 + (-0.900 + 0.433i)T^{2} \)
41 \( 1 + 0.445T + T^{2} \)
43 \( 1 + (0.193 + 0.400i)T + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (1.21 + 0.277i)T + (0.900 + 0.433i)T^{2} \)
53 \( 1 + (0.623 + 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-1.24 + 1.56i)T + (-0.222 - 0.974i)T^{2} \)
67 \( 1 + (-1.94 + 0.445i)T + (0.900 - 0.433i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.623 - 0.781i)T^{2} \)
79 \( 1 + (0.900 - 0.433i)T^{2} \)
83 \( 1 + (0.347 - 0.277i)T + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (-1.12 - 0.541i)T + (0.623 + 0.781i)T^{2} \)
97 \( 1 + (-0.222 + 0.974i)T^{2} \)
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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

−9.039217259152639321719998027302, −8.384125982917710726472815648722, −8.067318821266517844122067913306, −6.59120889475376402088728693786, −5.30166671953470389347857371701, −4.90590416586240065840426741640, −4.05471292553665756340076532317, −3.36501074795867452381383593591, −2.39892930211577739969874939942, −1.88295828266240226039512735279, 1.31707285129093296990747119073, 2.35405919125741307006087781437, 3.50957065576376206138311254589, 4.02445569858331545933826104098, 5.11827972562006560299310557487, 6.11493779604669005524509808102, 6.95163167218750951059073610215, 7.56950690332454644139890849871, 7.945187800214825318876263027239, 8.510704472516818529235998726660

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