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Properties

Label 2-30e2-20.19-c2-0-46
Degree $2$
Conductor $900$
Sign $0.591 + 0.806i$
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  + (−0.169 − 1.99i)2-s + (−3.94 + 0.675i)4-s + 12.3·7-s + (2.01 + 7.74i)8-s − 11.0i·11-s + 2.82i·13-s + (−2.10 − 24.7i)14-s + (15.0 − 5.32i)16-s + 6.52i·17-s + 27.9i·19-s + (−22.0 + 1.87i)22-s − 7.90·23-s + (5.61 − 0.477i)26-s + (−48.8 + 8.37i)28-s + 50.7·29-s + ⋯
L(s)  = 1  + (−0.0847 − 0.996i)2-s + (−0.985 + 0.168i)4-s + 1.77·7-s + (0.251 + 0.967i)8-s − 1.00i·11-s + 0.216i·13-s + (−0.150 − 1.76i)14-s + (0.942 − 0.332i)16-s + 0.383i·17-s + 1.47i·19-s + (−1.00 + 0.0850i)22-s − 0.343·23-s + (0.216 − 0.0183i)26-s + (−1.74 + 0.298i)28-s + 1.74·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.591 + 0.806i$
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: \((2,\ 900,\ (\ :1),\ 0.591 + 0.806i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.027720095\)
\(L(\frac12)\) \(\approx\) \(2.027720095\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.169 + 1.99i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 12.3T + 49T^{2} \)
11 \( 1 + 11.0iT - 121T^{2} \)
13 \( 1 - 2.82iT - 169T^{2} \)
17 \( 1 - 6.52iT - 289T^{2} \)
19 \( 1 - 27.9iT - 361T^{2} \)
23 \( 1 + 7.90T + 529T^{2} \)
29 \( 1 - 50.7T + 841T^{2} \)
31 \( 1 - 36.3iT - 961T^{2} \)
37 \( 1 - 18.9iT - 1.36e3T^{2} \)
41 \( 1 + 5.30T + 1.68e3T^{2} \)
43 \( 1 - 45.5T + 1.84e3T^{2} \)
47 \( 1 + 11.7T + 2.20e3T^{2} \)
53 \( 1 + 41.1iT - 2.80e3T^{2} \)
59 \( 1 - 10.7iT - 3.48e3T^{2} \)
61 \( 1 - 56.1T + 3.72e3T^{2} \)
67 \( 1 + 16.1T + 4.48e3T^{2} \)
71 \( 1 + 66.1iT - 5.04e3T^{2} \)
73 \( 1 - 15.6iT - 5.32e3T^{2} \)
79 \( 1 + 123. iT - 6.24e3T^{2} \)
83 \( 1 + 99.6T + 6.88e3T^{2} \)
89 \( 1 - 101.T + 7.92e3T^{2} \)
97 \( 1 + 127. iT - 9.40e3T^{2} \)
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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

−10.14793013822459239186667240723, −8.745555722083993367315964121654, −8.387305861122875945109920895345, −7.66684945166187840086105728680, −6.08709353285073408452420057039, −5.12056225122096555097192508822, −4.33383984608980915694063368371, −3.28844718317283130865224173782, −1.95352804442867363637546423225, −1.06980260750143217135890745169, 0.915022416472905920345129770646, 2.37677219822765580275356721615, 4.29181697142428871542522567822, 4.75516375114870954499488033362, 5.59396656842930350506439835917, 6.81402735964587291102406366344, 7.52438473870303379240285600377, 8.198713316166618130311710008914, 8.976196818485410999940255246121, 9.857918195328164874114586282559

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