Properties

Label 2-30e2-20.19-c2-0-79
Degree $2$
Conductor $900$
Sign $0.163 + 0.986i$
Analytic cond. $24.5232$
Root an. cond. $4.95209$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.90 − 0.618i)2-s + (3.23 − 2.35i)4-s + 5.25·7-s + (4.70 − 6.47i)8-s − 19.9i·11-s + 8.47i·13-s + (9.99 − 3.24i)14-s + (4.94 − 15.2i)16-s − 11.8i·17-s + 15.2i·19-s + (−12.3 − 37.8i)22-s + 0.555·23-s + (5.23 + 16.1i)26-s + (17.0 − 12.3i)28-s − 10.9·29-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + 0.751·7-s + (0.587 − 0.809i)8-s − 1.81i·11-s + 0.651i·13-s + (0.714 − 0.232i)14-s + (0.309 − 0.951i)16-s − 0.699i·17-s + 0.800i·19-s + (−0.559 − 1.72i)22-s + 0.0241·23-s + (0.201 + 0.619i)26-s + (0.607 − 0.441i)28-s − 0.377·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.163 + 0.986i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.163 + 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.779855940\)
\(L(\frac12)\) \(\approx\) \(3.779855940\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.90 + 0.618i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 5.25T + 49T^{2} \)
11 \( 1 + 19.9iT - 121T^{2} \)
13 \( 1 - 8.47iT - 169T^{2} \)
17 \( 1 + 11.8iT - 289T^{2} \)
19 \( 1 - 15.2iT - 361T^{2} \)
23 \( 1 - 0.555T + 529T^{2} \)
29 \( 1 + 10.9T + 841T^{2} \)
31 \( 1 + 8.29iT - 961T^{2} \)
37 \( 1 + 18.3iT - 1.36e3T^{2} \)
41 \( 1 - 14.5T + 1.68e3T^{2} \)
43 \( 1 - 22.2T + 1.84e3T^{2} \)
47 \( 1 - 53.3T + 2.20e3T^{2} \)
53 \( 1 + 66.3iT - 2.80e3T^{2} \)
59 \( 1 - 17.4iT - 3.48e3T^{2} \)
61 \( 1 - 90.1T + 3.72e3T^{2} \)
67 \( 1 + 50.2T + 4.48e3T^{2} \)
71 \( 1 - 80.7iT - 5.04e3T^{2} \)
73 \( 1 - 5.55iT - 5.32e3T^{2} \)
79 \( 1 - 13.8iT - 6.24e3T^{2} \)
83 \( 1 - 76.2T + 6.88e3T^{2} \)
89 \( 1 + 111.T + 7.92e3T^{2} \)
97 \( 1 + 92.8iT - 9.40e3T^{2} \)
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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

−9.914152352718175529950126013426, −8.878465570317507738376793037620, −7.975621320326049955620661430256, −7.00431669259741299257078399355, −5.95509516065862680020859155861, −5.37517300367809494157630871347, −4.25546868738986505374531154502, −3.40031763408424442854448448245, −2.23409898944043421477932975678, −0.925523787964879245609929429359, 1.66945626997710078208796103136, 2.68885360663636683419919397043, 4.07126010620538566974582117213, 4.76090063325721043735612083329, 5.56294086276630935004716217951, 6.68160140080564829861013407368, 7.45399530164737139664782558447, 8.088840597145716300610625249446, 9.213844501398420120631394238282, 10.33909230429730573478256448048

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