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Properties

Label 2-30e2-4.3-c2-0-70
Degree $2$
Conductor $900$
Sign $0.0876 + 0.996i$
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  + (−1.35 + 1.47i)2-s + (−0.350 − 3.98i)4-s − 10.9i·7-s + (6.35 + 4.86i)8-s − 11.3i·11-s + 10.8·13-s + (16.1 + 14.7i)14-s + (−15.7 + 2.79i)16-s + 15.8·17-s + 24.9i·19-s + (16.7 + 15.3i)22-s − 20.9i·23-s + (−14.5 + 15.9i)26-s + (−43.5 + 3.83i)28-s − 22.8·29-s + ⋯
L(s)  = 1  + (−0.675 + 0.737i)2-s + (−0.0876 − 0.996i)4-s − 1.55i·7-s + (0.793 + 0.608i)8-s − 1.03i·11-s + 0.831·13-s + (1.15 + 1.05i)14-s + (−0.984 + 0.174i)16-s + 0.929·17-s + 1.31i·19-s + (0.761 + 0.697i)22-s − 0.911i·23-s + (−0.561 + 0.613i)26-s + (−1.55 + 0.136i)28-s − 0.786·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0876 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0876 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.0876 + 0.996i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.0876 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.059908421\)
\(L(\frac12)\) \(\approx\) \(1.059908421\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 - 1.47i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 10.9iT - 49T^{2} \)
11 \( 1 + 11.3iT - 121T^{2} \)
13 \( 1 - 10.8T + 169T^{2} \)
17 \( 1 - 15.8T + 289T^{2} \)
19 \( 1 - 24.9iT - 361T^{2} \)
23 \( 1 + 20.9iT - 529T^{2} \)
29 \( 1 + 22.8T + 841T^{2} \)
31 \( 1 + 22.7iT - 961T^{2} \)
37 \( 1 - 19.1T + 1.36e3T^{2} \)
41 \( 1 + 17T + 1.68e3T^{2} \)
43 \( 1 + 6.51iT - 1.84e3T^{2} \)
47 \( 1 + 38.9iT - 2.20e3T^{2} \)
53 \( 1 - 13.2T + 2.80e3T^{2} \)
59 \( 1 - 95.6iT - 3.48e3T^{2} \)
61 \( 1 + 92.0T + 3.72e3T^{2} \)
67 \( 1 + 54.1iT - 4.48e3T^{2} \)
71 \( 1 + 68.5iT - 5.04e3T^{2} \)
73 \( 1 + 44.1T + 5.32e3T^{2} \)
79 \( 1 + 81.7iT - 6.24e3T^{2} \)
83 \( 1 + 27.9iT - 6.88e3T^{2} \)
89 \( 1 - 42.1T + 7.92e3T^{2} \)
97 \( 1 + 154.T + 9.40e3T^{2} \)
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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

−9.744554266329301959438849289547, −8.696199662130241357877181372749, −7.942036251674765478773051222882, −7.36080550015078404118224176405, −6.28311384288515579485858362231, −5.69961273517289797896915677424, −4.34865360572490819256462679803, −3.45771128528020383723799119484, −1.46277156697651199723226867098, −0.47604766490144949924899155153, 1.42965873428830152250188481583, 2.47848577391031902234105276002, 3.41467726964923526430106853775, 4.72650258506101924630394185181, 5.70976014716519054722620890248, 6.88962447955149451900525292301, 7.81039846154287743913695967678, 8.685594392896842642373810266813, 9.328086838243380662433019633488, 9.901459789179250772739508262264

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