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Properties

Label 2-30e2-9.5-c2-0-28
Degree $2$
Conductor $900$
Sign $-0.901 + 0.432i$
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  + (−1.83 − 2.37i)3-s + (4.13 + 7.15i)7-s + (−2.23 + 8.71i)9-s + (−1.19 + 0.687i)11-s + (−10.7 + 18.6i)13-s − 25.3i·17-s − 2.49·19-s + (9.36 − 22.9i)21-s + (−33.1 − 19.1i)23-s + (24.7 − 10.7i)27-s + (37.0 − 21.4i)29-s + (10.9 − 18.9i)31-s + (3.81 + 1.55i)33-s − 30.5·37-s + (64.0 − 8.78i)39-s + ⋯
L(s)  = 1  + (−0.612 − 0.790i)3-s + (0.590 + 1.02i)7-s + (−0.248 + 0.968i)9-s + (−0.108 + 0.0624i)11-s + (−0.829 + 1.43i)13-s − 1.48i·17-s − 0.131·19-s + (0.445 − 1.09i)21-s + (−1.44 − 0.833i)23-s + (0.917 − 0.397i)27-s + (1.27 − 0.738i)29-s + (0.353 − 0.611i)31-s + (0.115 + 0.0472i)33-s − 0.826·37-s + (1.64 − 0.225i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 + 0.432i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.901 + 0.432i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.901 + 0.432i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4413057823\)
\(L(\frac12)\) \(\approx\) \(0.4413057823\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.83 + 2.37i)T \)
5 \( 1 \)
good7 \( 1 + (-4.13 - 7.15i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (1.19 - 0.687i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (10.7 - 18.6i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 25.3iT - 289T^{2} \)
19 \( 1 + 2.49T + 361T^{2} \)
23 \( 1 + (33.1 + 19.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-37.0 + 21.4i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-10.9 + 18.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 30.5T + 1.36e3T^{2} \)
41 \( 1 + (-7.29 - 4.21i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (35.5 + 61.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (42.1 - 24.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 1.96iT - 2.80e3T^{2} \)
59 \( 1 + (-3.77 - 2.18i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (18.4 + 32.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-8.06 + 13.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 71.5iT - 5.04e3T^{2} \)
73 \( 1 + 122.T + 5.32e3T^{2} \)
79 \( 1 + (-3.98 - 6.90i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-90.2 + 52.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 9.37iT - 7.92e3T^{2} \)
97 \( 1 + (-6.86 - 11.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.531995218213025104808329289413, −8.598155524917036285518560033464, −7.82515674091833552603807267964, −6.88569631835383590868305530763, −6.19126854839140529497440012495, −5.12078456951732558669476679389, −4.51224487518580038237543600551, −2.54865899496376102497989489374, −1.90503379047814217801359561709, −0.15865701940722067656685213884, 1.30979570412542232870978600190, 3.15431036749123223751995438678, 4.10284715536963997002441566338, 4.95308436929042938346737115247, 5.78275999532451242185459132503, 6.77192958533137620427097933552, 7.88469537759494184996929324902, 8.455120911724193244001919943716, 9.852485201749622625117278297135, 10.34534814330692302206470175549

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