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Properties

Label 2-450-9.2-c2-0-11
Degree $2$
Conductor $450$
Sign $-0.00922 - 0.999i$
Analytic cond. $12.2616$
Root an. cond. $3.50165$
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  + (1.22 + 0.707i)2-s + (−2.44 + 1.73i)3-s + (0.999 + 1.73i)4-s + (−4.22 + 0.389i)6-s + (3.17 − 5.49i)7-s + 2.82i·8-s + (2.99 − 8.48i)9-s + (8.17 + 4.71i)11-s + (−5.44 − 2.51i)12-s + (9.84 + 17.0i)13-s + (7.77 − 4.48i)14-s + (−2.00 + 3.46i)16-s + 1.90i·17-s + (9.67 − 8.27i)18-s + 4.69·19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.816 + 0.577i)3-s + (0.249 + 0.433i)4-s + (−0.704 + 0.0648i)6-s + (0.453 − 0.785i)7-s + 0.353i·8-s + (0.333 − 0.942i)9-s + (0.743 + 0.429i)11-s + (−0.454 − 0.209i)12-s + (0.757 + 1.31i)13-s + (0.555 − 0.320i)14-s + (−0.125 + 0.216i)16-s + 0.112i·17-s + (0.537 − 0.459i)18-s + 0.247·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00922 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.00922 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.00922 - 0.999i$
Analytic conductor: \(12.2616\)
Root analytic conductor: \(3.50165\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1),\ -0.00922 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.027542705\)
\(L(\frac12)\) \(\approx\) \(2.027542705\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 + (2.44 - 1.73i)T \)
5 \( 1 \)
good7 \( 1 + (-3.17 + 5.49i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-8.17 - 4.71i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-9.84 - 17.0i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 1.90iT - 289T^{2} \)
19 \( 1 - 4.69T + 361T^{2} \)
23 \( 1 + (8.17 - 4.71i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (2.84 + 1.64i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-20.5 - 35.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 17.3T + 1.36e3T^{2} \)
41 \( 1 + (53.5 - 30.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-0.477 + 0.826i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-12.2 - 7.05i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 9.53iT - 2.80e3T^{2} \)
59 \( 1 + (-79.2 + 45.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-37.5 + 65.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-15.4 - 26.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 85.9iT - 5.04e3T^{2} \)
73 \( 1 - 96.0T + 5.32e3T^{2} \)
79 \( 1 + (14.8 - 25.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-76.1 - 43.9i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 41.3iT - 7.92e3T^{2} \)
97 \( 1 + (-47.9 + 83.0i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.34399013348986047481904036817, −10.35759099570765820841022328276, −9.416332360491096018788192194961, −8.341396549002316989717883630724, −6.93428324872963338170673153307, −6.52888991105375413515133053875, −5.25191913101009285973581735384, −4.33351358563758517051952525070, −3.67671837555965902666453376239, −1.44012242179938115738635517860, 0.876263518207804846469554753892, 2.25539259001146811665810036940, 3.70949498728045957766961427209, 5.12507115019193949456204310419, 5.77320652187322000761771959248, 6.59194322108848878462994657230, 7.85163691183019772464600475369, 8.743280569702237127370339656592, 10.12361231443348872382669704366, 10.91441083960441478339431832145

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