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Properties

Label 2-30e2-45.4-c1-0-11
Degree $2$
Conductor $900$
Sign $-0.0634 + 0.997i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.39 + 1.02i)3-s + (−0.589 + 0.340i)7-s + (0.880 − 2.86i)9-s + (0.840 + 1.45i)11-s + (−4.45 − 2.57i)13-s + 1.31i·17-s + 0.324·19-s + (0.470 − 1.08i)21-s + (−3.28 − 1.89i)23-s + (1.72 + 4.90i)27-s + (−4.32 − 7.48i)29-s + (2.07 − 3.58i)31-s + (−2.66 − 1.16i)33-s − 1.35i·37-s + (8.85 − 1.00i)39-s + ⋯
L(s)  = 1  + (−0.804 + 0.594i)3-s + (−0.222 + 0.128i)7-s + (0.293 − 0.955i)9-s + (0.253 + 0.438i)11-s + (−1.23 − 0.713i)13-s + 0.320i·17-s + 0.0744·19-s + (0.102 − 0.235i)21-s + (−0.684 − 0.395i)23-s + (0.331 + 0.943i)27-s + (−0.802 − 1.39i)29-s + (0.372 − 0.644i)31-s + (−0.464 − 0.202i)33-s − 0.222i·37-s + (1.41 − 0.160i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0634 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0634 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.0634 + 0.997i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.0634 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.362300 - 0.386075i\)
\(L(\frac12)\) \(\approx\) \(0.362300 - 0.386075i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.39 - 1.02i)T \)
5 \( 1 \)
good7 \( 1 + (0.589 - 0.340i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.840 - 1.45i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.45 + 2.57i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.31iT - 17T^{2} \)
19 \( 1 - 0.324T + 19T^{2} \)
23 \( 1 + (3.28 + 1.89i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.32 + 7.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.07 + 3.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.35iT - 37T^{2} \)
41 \( 1 + (-3.57 + 6.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.31 - 3.64i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-11.2 + 6.48i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.83iT - 53T^{2} \)
59 \( 1 + (-4.40 + 7.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.98 + 8.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.61 + 2.08i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.891T + 71T^{2} \)
73 \( 1 + 7.82iT - 73T^{2} \)
79 \( 1 + (-4.82 - 8.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.20 + 2.42i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 17.4T + 89T^{2} \)
97 \( 1 + (7.73 - 4.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.829605769014064736051085021859, −9.470668851223588077931417179816, −8.171704798901901478089555379527, −7.26262575270267778869206097516, −6.28255827462128474927178007504, −5.50493848189240648364986129417, −4.59123896954899046290094801897, −3.69163537615172258322502960662, −2.26880476721837645093085993245, −0.28426954046825259388422264654, 1.40975472677703276682665329604, 2.74630430089706387538488360930, 4.21777757421264259834775619809, 5.17982068415628810145486751456, 6.03132905396203482340660300398, 7.01395698440087209287223552111, 7.46325547039488840265898041724, 8.644988656338184844667715191980, 9.588594182241647016104659686851, 10.40218913475436362677543349056

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