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Properties

Label 2-30e2-20.19-c2-0-68
Degree $2$
Conductor $900$
Sign $-0.631 + 0.775i$
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  + (−1.08 + 1.67i)2-s + (−1.64 − 3.64i)4-s − 0.596·7-s + (7.91 + 1.19i)8-s + 9.27i·11-s − 23.5i·13-s + (0.647 − 1.00i)14-s + (−10.5 + 11.9i)16-s + 3.97i·17-s − 7.04i·19-s + (−15.5 − 10.0i)22-s − 32.0·23-s + (39.4 + 25.5i)26-s + (0.980 + 2.17i)28-s + 35.6·29-s + ⋯
L(s)  = 1  + (−0.542 + 0.839i)2-s + (−0.410 − 0.911i)4-s − 0.0852·7-s + (0.988 + 0.149i)8-s + 0.843i·11-s − 1.80i·13-s + (0.0462 − 0.0715i)14-s + (−0.662 + 0.749i)16-s + 0.233i·17-s − 0.370i·19-s + (−0.708 − 0.457i)22-s − 1.39·23-s + (1.51 + 0.981i)26-s + (0.0350 + 0.0776i)28-s + 1.23·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.631 + 0.775i$
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: \((2,\ 900,\ (\ :1),\ -0.631 + 0.775i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.09487269260\)
\(L(\frac12)\) \(\approx\) \(0.09487269260\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 - 1.67i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 0.596T + 49T^{2} \)
11 \( 1 - 9.27iT - 121T^{2} \)
13 \( 1 + 23.5iT - 169T^{2} \)
17 \( 1 - 3.97iT - 289T^{2} \)
19 \( 1 + 7.04iT - 361T^{2} \)
23 \( 1 + 32.0T + 529T^{2} \)
29 \( 1 - 35.6T + 841T^{2} \)
31 \( 1 - 59.2iT - 961T^{2} \)
37 \( 1 - 5.38iT - 1.36e3T^{2} \)
41 \( 1 + 40.0T + 1.68e3T^{2} \)
43 \( 1 - 36.1T + 1.84e3T^{2} \)
47 \( 1 + 74.0T + 2.20e3T^{2} \)
53 \( 1 - 2.55iT - 2.80e3T^{2} \)
59 \( 1 + 36.4iT - 3.48e3T^{2} \)
61 \( 1 + 8.73T + 3.72e3T^{2} \)
67 \( 1 + 69.7T + 4.48e3T^{2} \)
71 \( 1 + 59.2iT - 5.04e3T^{2} \)
73 \( 1 + 83.0iT - 5.32e3T^{2} \)
79 \( 1 - 65.8iT - 6.24e3T^{2} \)
83 \( 1 + 129.T + 6.88e3T^{2} \)
89 \( 1 + 130.T + 7.92e3T^{2} \)
97 \( 1 + 93.1iT - 9.40e3T^{2} \)
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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

−9.709928857685600053117745973090, −8.477146649998528162540234813374, −8.051058084338464327508956910572, −7.09012115638702308680692764597, −6.28500043829831675368949744665, −5.32865167274491236079519143484, −4.55384239997356022332112783459, −3.08641039368152982470335092209, −1.52613509498636396686310764277, −0.03788805528801830638275087112, 1.51381992932725908376413306316, 2.59485078362432427402103471065, 3.82359103040222804525452399015, 4.52369077096558330155479554690, 5.98480631625734256201280806380, 6.90848579058451005154248796663, 7.999136494251803497485008407956, 8.626244066964250803201202863726, 9.551758437006021630547863162522, 10.05151909011515868613603083786

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