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Properties

Label 2-30e2-15.2-c3-0-17
Degree $2$
Conductor $900$
Sign $-0.986 - 0.161i$
Analytic cond. $53.1017$
Root an. cond. $7.28709$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−17.1 − 17.1i)7-s − 68.0i·11-s + (33.1 − 33.1i)13-s + (−15.5 + 15.5i)17-s − 40.1i·19-s + (52.1 + 52.1i)23-s + 97.9·29-s − 206.·31-s + (238. + 238. i)37-s + 43.1i·41-s + (−255. + 255. i)43-s + (−134. + 134. i)47-s + 248. i·49-s + (−458. − 458. i)53-s − 748.·59-s + ⋯
L(s)  = 1  + (−0.928 − 0.928i)7-s − 1.86i·11-s + (0.707 − 0.707i)13-s + (−0.222 + 0.222i)17-s − 0.484i·19-s + (0.473 + 0.473i)23-s + 0.627·29-s − 1.19·31-s + (1.05 + 1.05i)37-s + 0.164i·41-s + (−0.905 + 0.905i)43-s + (−0.418 + 0.418i)47-s + 0.724i·49-s + (−1.18 − 1.18i)53-s − 1.65·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.986 - 0.161i$
Analytic conductor: \(53.1017\)
Root analytic conductor: \(7.28709\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :3/2),\ -0.986 - 0.161i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7252689384\)
\(L(\frac12)\) \(\approx\) \(0.7252689384\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 + (17.1 + 17.1i)T + 343iT^{2} \)
11 \( 1 + 68.0iT - 1.33e3T^{2} \)
13 \( 1 + (-33.1 + 33.1i)T - 2.19e3iT^{2} \)
17 \( 1 + (15.5 - 15.5i)T - 4.91e3iT^{2} \)
19 \( 1 + 40.1iT - 6.85e3T^{2} \)
23 \( 1 + (-52.1 - 52.1i)T + 1.21e4iT^{2} \)
29 \( 1 - 97.9T + 2.43e4T^{2} \)
31 \( 1 + 206.T + 2.97e4T^{2} \)
37 \( 1 + (-238. - 238. i)T + 5.06e4iT^{2} \)
41 \( 1 - 43.1iT - 6.89e4T^{2} \)
43 \( 1 + (255. - 255. i)T - 7.95e4iT^{2} \)
47 \( 1 + (134. - 134. i)T - 1.03e5iT^{2} \)
53 \( 1 + (458. + 458. i)T + 1.48e5iT^{2} \)
59 \( 1 + 748.T + 2.05e5T^{2} \)
61 \( 1 - 240.T + 2.26e5T^{2} \)
67 \( 1 + (133. + 133. i)T + 3.00e5iT^{2} \)
71 \( 1 + 899. iT - 3.57e5T^{2} \)
73 \( 1 + (26.5 - 26.5i)T - 3.89e5iT^{2} \)
79 \( 1 - 112. iT - 4.93e5T^{2} \)
83 \( 1 + (644. + 644. i)T + 5.71e5iT^{2} \)
89 \( 1 - 554.T + 7.04e5T^{2} \)
97 \( 1 + (-1.24e3 - 1.24e3i)T + 9.12e5iT^{2} \)
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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

−9.284945161617975574299712941166, −8.428358172347610281232858781101, −7.66762059864145797301416535889, −6.47930086958755028800474743740, −6.05643595494048267243391826560, −4.82315684866160162344080637909, −3.45251098295780027560873215492, −3.15828481397898227107454719676, −1.15685524573008074933237484902, −0.20029059867829206105051361313, 1.70032246664575322560969828214, 2.65444924713947472197257396130, 3.90242906659273310501323475729, 4.85238395417734512526168988865, 5.94585742060679880344260336712, 6.73091833047101448598294376883, 7.48733491809741411799276656207, 8.726564689760703294054116973020, 9.339640786661826042343335417998, 9.968786781233269479707333936115

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