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Properties

Label 2-2900-116.71-c0-0-1
Degree $2$
Conductor $2900$
Sign $-0.973 - 0.230i$
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  + (0.900 + 0.433i)2-s + (−1.12 + 1.40i)3-s + (0.623 + 0.781i)4-s + (−1.62 + 0.781i)6-s + (1.22 + 0.974i)7-s + (0.222 + 0.974i)8-s + (−0.500 − 2.19i)9-s − 1.80·12-s + (0.678 + 1.40i)14-s + (−0.222 + 0.974i)16-s + (0.499 − 2.19i)18-s + (−2.74 + 0.626i)21-s + (0.376 + 0.781i)23-s + (−1.62 − 0.781i)24-s + (2.02 + 0.974i)27-s + 1.56i·28-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (−1.12 + 1.40i)3-s + (0.623 + 0.781i)4-s + (−1.62 + 0.781i)6-s + (1.22 + 0.974i)7-s + (0.222 + 0.974i)8-s + (−0.500 − 2.19i)9-s − 1.80·12-s + (0.678 + 1.40i)14-s + (−0.222 + 0.974i)16-s + (0.499 − 2.19i)18-s + (−2.74 + 0.626i)21-s + (0.376 + 0.781i)23-s + (−1.62 − 0.781i)24-s + (2.02 + 0.974i)27-s + 1.56i·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $-0.973 - 0.230i$
Analytic conductor: \(1.44728\)
Root analytic conductor: \(1.20303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2900,\ (\ :0),\ -0.973 - 0.230i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.623182743\)
\(L(\frac12)\) \(\approx\) \(1.623182743\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 \)
29 \( 1 + (0.623 - 0.781i)T \)
good3 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
7 \( 1 + (-1.22 - 0.974i)T + (0.222 + 0.974i)T^{2} \)
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 \( 1 + (-0.376 - 0.781i)T + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.623 + 0.781i)T^{2} \)
37 \( 1 + (0.900 - 0.433i)T^{2} \)
41 \( 1 + 1.94iT - T^{2} \)
43 \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.277 + 1.21i)T + (-0.900 - 0.433i)T^{2} \)
53 \( 1 + (0.623 + 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.222 + 0.974i)T^{2} \)
67 \( 1 + (0.900 - 0.433i)T^{2} \)
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 \( 1 + (-1.52 + 1.21i)T + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.678 + 1.40i)T + (-0.623 - 0.781i)T^{2} \)
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

−9.130743761978035103556613336015, −8.704224957938148303509089600396, −7.66319991519794206231580588611, −6.75483175062301706517001354829, −5.75068409712510847900123452045, −5.37951825088889665920869735804, −4.88348483544199306205855004570, −4.05782444279145603161463863814, −3.26715495632590610333719351009, −1.93106531736530508530601325435, 0.928209148384403253963107862404, 1.64374143025451386380059229811, 2.62526429886424828190446742306, 4.10014169725682306978652134229, 4.82715301405186884734537690833, 5.44780023103421837641255296844, 6.36706580166310990271399517388, 6.82851532716987329687244665222, 7.70488115705777218801941125574, 8.077005906528276706823393818767

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