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Properties

Label 2-450-45.34-c1-0-7
Degree $2$
Conductor $450$
Sign $0.958 - 0.285i$
Analytic cond. $3.59326$
Root an. cond. $1.89559$
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  + (0.866 + 0.5i)2-s + (−0.866 − 1.5i)3-s + (0.499 + 0.866i)4-s − 1.73i·6-s + (3.46 + 2i)7-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + (−1.5 + 2.59i)11-s + (0.866 − 1.49i)12-s + (3.46 − 2i)13-s + (1.99 + 3.46i)14-s + (−0.5 + 0.866i)16-s − 3i·17-s + (−2.59 + 1.5i)18-s + 4·19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 − 0.866i)3-s + (0.249 + 0.433i)4-s − 0.707i·6-s + (1.30 + 0.755i)7-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.452 + 0.783i)11-s + (0.250 − 0.433i)12-s + (0.960 − 0.554i)13-s + (0.534 + 0.925i)14-s + (−0.125 + 0.216i)16-s − 0.727i·17-s + (−0.612 + 0.353i)18-s + 0.917·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.958 - 0.285i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.958 - 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88282 + 0.274088i\)
\(L(\frac12)\) \(\approx\) \(1.88282 + 0.274088i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 \)
good7 \( 1 + (-3.46 - 2i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.46 + 2i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (10.3 + 6i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.46 - 2i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 14iT - 73T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.79 + 4.5i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (-6.06 - 3.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.45989072443151541653362082058, −10.63961700627426411747550848195, −9.056137210567219254300892722655, −8.028659385996726125242590012922, −7.48156421147113461603827302648, −6.37442517262936953245878493036, −5.28653110991761233435234100526, −4.87081096388530005545633985034, −2.92870463312217593216902191729, −1.60666082521912135561151358721, 1.32838530475066507362681787699, 3.36143391532898768042897988904, 4.20401803572603937524063952714, 5.19056635732825254653682358924, 5.93456960524839809523389131597, 7.27748430076572987679496070126, 8.445499737933950675564880653681, 9.418493648345302512799056367286, 10.64180814303195327167331279225, 11.07114830009192843448431893885

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