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Properties

Label 2-30e2-180.119-c1-0-101
Degree $2$
Conductor $900$
Sign $-0.885 + 0.464i$
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  + (1.09 − 0.896i)2-s + (0.654 − 1.60i)3-s + (0.394 − 1.96i)4-s + (−0.721 − 2.34i)6-s + (−0.604 + 1.04i)7-s + (−1.32 − 2.49i)8-s + (−2.14 − 2.09i)9-s + (1.52 − 2.63i)11-s + (−2.88 − 1.91i)12-s + (4.39 − 2.53i)13-s + (0.276 + 1.68i)14-s + (−3.68 − 1.54i)16-s − 2.23·17-s + (−4.22 − 0.373i)18-s + 2.53i·19-s + ⋯
L(s)  = 1  + (0.773 − 0.633i)2-s + (0.377 − 0.925i)3-s + (0.197 − 0.980i)4-s + (−0.294 − 0.955i)6-s + (−0.228 + 0.395i)7-s + (−0.468 − 0.883i)8-s + (−0.714 − 0.699i)9-s + (0.458 − 0.794i)11-s + (−0.833 − 0.552i)12-s + (1.21 − 0.704i)13-s + (0.0739 + 0.450i)14-s + (−0.922 − 0.386i)16-s − 0.543·17-s + (−0.996 − 0.0881i)18-s + 0.582i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.885 + 0.464i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.885 + 0.464i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.606310 - 2.45959i\)
\(L(\frac12)\) \(\approx\) \(0.606310 - 2.45959i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.09 + 0.896i)T \)
3 \( 1 + (-0.654 + 1.60i)T \)
5 \( 1 \)
good7 \( 1 + (0.604 - 1.04i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.52 + 2.63i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.39 + 2.53i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.23T + 17T^{2} \)
19 \( 1 - 2.53iT - 19T^{2} \)
23 \( 1 + (6.02 - 3.48i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.84 - 2.79i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.83 + 1.63i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.3iT - 37T^{2} \)
41 \( 1 + (-3.70 + 2.13i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.15 - 3.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.92 + 5.15i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.42T + 53T^{2} \)
59 \( 1 + (-4.46 - 7.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.32 + 2.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.20 - 12.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.49T + 71T^{2} \)
73 \( 1 + 5.10iT - 73T^{2} \)
79 \( 1 + (-3.26 - 1.88i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.495 - 0.286i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.40iT - 89T^{2} \)
97 \( 1 + (-10.0 - 5.82i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.836582565824332068220472781649, −8.838186094180537492408376159474, −8.218125529778877043968631960165, −6.96916501710733264490952412135, −5.99008283035843631873435850957, −5.71720836592465518823619005163, −3.97958091130874354647319022507, −3.25933986019050317577416939719, −2.14876497387142731118056700472, −0.925734019356732656459603458049, 2.27843204991517502455301284217, 3.54295316964289743308750421023, 4.28075773193103444505023128699, 4.90116776692186646264114051504, 6.29042749583832919557483354231, 6.73516935273070240569684251957, 8.075484924290139156392239199755, 8.614229178633410754460810519204, 9.564468089954326193874438652667, 10.40693743097555592783812849112

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