Properties

Label 2-450-25.14-c1-0-3
Degree $2$
Conductor $450$
Sign $-0.785 - 0.618i$
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  + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s + (−0.530 + 2.17i)5-s − 0.273i·7-s + (−0.951 + 0.309i)8-s + (−2.06 + 0.847i)10-s + (−3.87 + 2.81i)11-s + (−1.01 + 1.40i)13-s + (0.221 − 0.160i)14-s + (−0.809 − 0.587i)16-s + (1.79 − 0.584i)17-s + (0.930 + 2.86i)19-s + (−1.90 − 1.17i)20-s + (−4.55 − 1.48i)22-s + (2.73 + 3.75i)23-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (−0.154 + 0.475i)4-s + (−0.237 + 0.971i)5-s − 0.103i·7-s + (−0.336 + 0.109i)8-s + (−0.654 + 0.268i)10-s + (−1.16 + 0.849i)11-s + (−0.282 + 0.388i)13-s + (0.0591 − 0.0429i)14-s + (−0.202 − 0.146i)16-s + (0.436 − 0.141i)17-s + (0.213 + 0.656i)19-s + (−0.425 − 0.262i)20-s + (−0.971 − 0.315i)22-s + (0.569 + 0.783i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 - 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.785 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.785 - 0.618i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.785 - 0.618i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.430026 + 1.24061i\)
\(L(\frac12)\) \(\approx\) \(0.430026 + 1.24061i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 \)
5 \( 1 + (0.530 - 2.17i)T \)
good7 \( 1 + 0.273iT - 7T^{2} \)
11 \( 1 + (3.87 - 2.81i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.01 - 1.40i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.79 + 0.584i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.930 - 2.86i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.73 - 3.75i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.80 + 8.64i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.18 - 3.64i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.29 - 7.28i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.13 + 4.45i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.88iT - 43T^{2} \)
47 \( 1 + (-1.12 - 0.367i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-11.0 - 3.58i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-11.5 - 8.39i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.16 + 3.02i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-5.98 + 1.94i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-0.433 + 1.33i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.04 + 1.44i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.96 + 15.2i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (14.3 - 4.67i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-11.9 + 8.71i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-5.59 - 1.81i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.65447041093236129533649429691, −10.35980912393881506146819892526, −9.906841200038806953336656165195, −8.440605102179599249480834352950, −7.49246945294152952787156894967, −6.99668257701268417610338979109, −5.80927758112910908725244176948, −4.80033483738633088175355139718, −3.58567103028615275454632028059, −2.42441790941694410586932354657, 0.71487826075278723959125970772, 2.53800543652310987743690125520, 3.72667450844573541060004271184, 5.10944632453876424860889940540, 5.45920759201075446586067790435, 7.02758964057884614611459626406, 8.279801399194792217824699427267, 8.846170245933919037478475273889, 10.04160469596340021191024394960, 10.79786127975604242058446545128

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