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Properties

Label 2-450-225.106-c1-0-21
Degree $2$
Conductor $450$
Sign $0.515 + 0.856i$
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  + (−0.978 − 0.207i)2-s + (1.35 − 1.08i)3-s + (0.913 + 0.406i)4-s + (2.22 + 0.190i)5-s + (−1.54 + 0.775i)6-s + (0.821 − 1.42i)7-s + (−0.809 − 0.587i)8-s + (0.664 − 2.92i)9-s + (−2.13 − 0.649i)10-s + (−0.250 − 0.0532i)11-s + (1.67 − 0.436i)12-s + (0.115 − 0.0244i)13-s + (−1.09 + 1.22i)14-s + (3.22 − 2.14i)15-s + (0.669 + 0.743i)16-s + (−2.89 − 2.10i)17-s + ⋯
L(s)  = 1  + (−0.691 − 0.147i)2-s + (0.781 − 0.623i)3-s + (0.456 + 0.203i)4-s + (0.996 + 0.0851i)5-s + (−0.632 + 0.316i)6-s + (0.310 − 0.537i)7-s + (−0.286 − 0.207i)8-s + (0.221 − 0.975i)9-s + (−0.676 − 0.205i)10-s + (−0.0755 − 0.0160i)11-s + (0.483 − 0.126i)12-s + (0.0319 − 0.00678i)13-s + (−0.293 + 0.326i)14-s + (0.831 − 0.555i)15-s + (0.167 + 0.185i)16-s + (−0.702 − 0.510i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.515 + 0.856i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.515 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37115 - 0.775314i\)
\(L(\frac12)\) \(\approx\) \(1.37115 - 0.775314i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.978 + 0.207i)T \)
3 \( 1 + (-1.35 + 1.08i)T \)
5 \( 1 + (-2.22 - 0.190i)T \)
good7 \( 1 + (-0.821 + 1.42i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.250 + 0.0532i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-0.115 + 0.0244i)T + (11.8 - 5.28i)T^{2} \)
17 \( 1 + (2.89 + 2.10i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.33 - 2.42i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (3.07 - 3.41i)T + (-2.40 - 22.8i)T^{2} \)
29 \( 1 + (-0.658 - 6.26i)T + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (-0.226 + 2.15i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-0.176 + 0.543i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (11.8 - 2.52i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (-6.48 + 11.2i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.165 + 1.57i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (-2.18 + 1.58i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.69 + 0.573i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (5.99 + 1.27i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (-0.341 + 3.24i)T + (-65.5 - 13.9i)T^{2} \)
71 \( 1 + (-4.88 + 3.55i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.40 - 13.5i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.23 - 11.7i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (4.39 - 1.95i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (-0.755 - 2.32i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-0.609 - 5.79i)T + (-94.8 + 20.1i)T^{2} \)
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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

−10.72672504113251867825187720819, −9.863248814472623987302729290710, −9.175113564139204074234306419640, −8.296128708077739587264931996271, −7.33144060912926424997756945918, −6.64166043028697512611383603620, −5.39185114825646572157463385616, −3.66175953928193589075170930853, −2.39112781122811123324521053755, −1.31820937588948890191444632586, 1.87452636730406476301854323398, 2.83818557617555024921362555304, 4.52112425161111675328364876266, 5.58692613117210011425025010057, 6.64811659028290245297090685674, 7.934086221596306715967413112336, 8.694932419172270576796168245137, 9.370617989167423711742566497896, 10.12191145303470264738917006280, 10.85466179376680644894571992900

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