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Properties

Label 2-450-45.38-c1-0-0
Degree $2$
Conductor $450$
Sign $-0.987 + 0.157i$
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  + (−0.965 + 0.258i)2-s + (−0.448 + 1.67i)3-s + (0.866 − 0.499i)4-s − 1.73i·6-s + (−0.707 + 0.707i)8-s + (−2.59 − 1.50i)9-s + (−1.5 − 0.866i)11-s + (0.448 + 1.67i)12-s + (−1.79 + 6.69i)13-s + (0.500 − 0.866i)16-s + (−2.12 − 2.12i)17-s + (2.89 + 0.776i)18-s + 4i·19-s + (1.67 + 0.448i)22-s + (−5.79 − 1.55i)23-s + (−0.866 − 1.5i)24-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.258 + 0.965i)3-s + (0.433 − 0.249i)4-s − 0.707i·6-s + (−0.249 + 0.249i)8-s + (−0.866 − 0.5i)9-s + (−0.452 − 0.261i)11-s + (0.129 + 0.482i)12-s + (−0.497 + 1.85i)13-s + (0.125 − 0.216i)16-s + (−0.514 − 0.514i)17-s + (0.683 + 0.183i)18-s + 0.917i·19-s + (0.356 + 0.0955i)22-s + (−1.20 − 0.323i)23-s + (−0.176 − 0.306i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.987 + 0.157i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.987 + 0.157i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0293494 - 0.370810i\)
\(L(\frac12)\) \(\approx\) \(0.0293494 - 0.370810i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (0.448 - 1.67i)T \)
5 \( 1 \)
good7 \( 1 + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.79 - 6.69i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (2.12 + 2.12i)T + 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (5.79 + 1.55i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.73 - 3i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.89 + 4.89i)T - 37iT^{2} \)
41 \( 1 + (6 - 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.36 - 2.24i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (11.5 - 3.10i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \)
59 \( 1 + (-4.33 - 7.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.34 + 0.896i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (-4.89 - 4.89i)T + 73iT^{2} \)
79 \( 1 + (-3.46 - 2i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.32 + 8.69i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 1.73T + 89T^{2} \)
97 \( 1 + (-4.03 - 15.0i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.56803829445336250964927502251, −10.43549859587062970300215326332, −9.757501593360057240854732655709, −9.031474426058974411388790547176, −8.146620631210004043613257020081, −6.92415328007068817415001112276, −5.99771608654489794179336355451, −4.86284770496604200918936157666, −3.80459739735309075213788624186, −2.18159339764586540696604536865, 0.27706917331607085837990708046, 1.99624713374154052971030551586, 3.13986599142439314715877444946, 5.06297551065173506716797755046, 6.07067206631636662109431199244, 7.12449929693604420597553558793, 7.931012877172357612957419450317, 8.530427671675877515847950416753, 9.886360444586240747620266785421, 10.58597999614349860647563539504

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