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Properties

Label 2-30e2-20.19-c2-0-32
Degree $2$
Conductor $900$
Sign $-0.0218 - 0.999i$
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  + (1.95 + 0.438i)2-s + (3.61 + 1.71i)4-s − 6.33·7-s + (6.30 + 4.92i)8-s − 9.27i·11-s + 18.5i·13-s + (−12.3 − 2.77i)14-s + (10.1 + 12.3i)16-s + 13.9i·17-s + 17.2i·19-s + (4.06 − 18.1i)22-s + 33.7·23-s + (−8.13 + 36.2i)26-s + (−22.8 − 10.8i)28-s − 28.6·29-s + ⋯
L(s)  = 1  + (0.975 + 0.219i)2-s + (0.904 + 0.427i)4-s − 0.904·7-s + (0.788 + 0.615i)8-s − 0.843i·11-s + 1.42i·13-s + (−0.882 − 0.198i)14-s + (0.634 + 0.772i)16-s + 0.818i·17-s + 0.907i·19-s + (0.184 − 0.823i)22-s + 1.46·23-s + (−0.312 + 1.39i)26-s + (−0.817 − 0.386i)28-s − 0.986·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0218 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0218 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.0218 - 0.999i$
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: \((2,\ 900,\ (\ :1),\ -0.0218 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.995362097\)
\(L(\frac12)\) \(\approx\) \(2.995362097\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.95 - 0.438i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 6.33T + 49T^{2} \)
11 \( 1 + 9.27iT - 121T^{2} \)
13 \( 1 - 18.5iT - 169T^{2} \)
17 \( 1 - 13.9iT - 289T^{2} \)
19 \( 1 - 17.2iT - 361T^{2} \)
23 \( 1 - 33.7T + 529T^{2} \)
29 \( 1 + 28.6T + 841T^{2} \)
31 \( 1 - 23.4iT - 961T^{2} \)
37 \( 1 - 67.3iT - 1.36e3T^{2} \)
41 \( 1 - 44.0T + 1.68e3T^{2} \)
43 \( 1 - 50.2T + 1.84e3T^{2} \)
47 \( 1 + 31.1T + 2.20e3T^{2} \)
53 \( 1 + 81.6iT - 2.80e3T^{2} \)
59 \( 1 - 19.2iT - 3.48e3T^{2} \)
61 \( 1 + 53.1T + 3.72e3T^{2} \)
67 \( 1 - 4.49T + 4.48e3T^{2} \)
71 \( 1 + 13.3iT - 5.04e3T^{2} \)
73 \( 1 - 40.8iT - 5.32e3T^{2} \)
79 \( 1 + 141. iT - 6.24e3T^{2} \)
83 \( 1 + 69.8T + 6.88e3T^{2} \)
89 \( 1 + 46.3T + 7.92e3T^{2} \)
97 \( 1 + 68.5iT - 9.40e3T^{2} \)
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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

−10.27670631655100240723293156689, −9.227034502516910793335846009632, −8.385764485711257115325364157188, −7.29417226899458528487790462490, −6.47118921655418465376544751100, −5.92889233679953643692280398543, −4.77762116958149308447027613778, −3.75570053318088581991282075932, −3.02660404164275013913206394622, −1.60074855774131845202938647051, 0.69145137482740411076703516606, 2.46069808909722573613742220891, 3.17120724407889808356150154010, 4.31423003287712208978196877063, 5.28069536527382403734780117770, 6.02634521502165028359896317971, 7.16142130643116385588210952415, 7.53606948984374903806757439629, 9.193494497818290654108477591442, 9.728844728720775853080343468178

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