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Properties

Label 2-2900-116.71-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(0.8203505328\)
\(L(\frac12)\) \(\approx\) \(0.8203505328\)
\(L(1)\) 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

−8.635530668861825149347379061063, −7.71631659474979089270191232492, −7.24647733556127697093508103719, −6.72538296674570081435097385402, −6.01620724902509680094688820606, −4.04758385566452550283155090221, −3.35832313044318672217616394573, −2.63862046865577423021281237932, −1.68614632818631866048798622616, −0.56588709249792669385675785002, 2.07103239531292337907656771852, 2.87971760283391037280769453319, 3.59116530505233607060810097080, 4.72497757958544543489325474999, 5.63525232757549284063821070004, 6.26814710128815253714114809852, 7.37345616482938446674036269615, 8.171761259561142487315068049673, 8.751681659963585228071702819453, 9.413443092505895858005462040617

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