Properties

Label 2-450-25.14-c1-0-6
Degree $2$
Conductor $450$
Sign $0.959 + 0.280i$
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 + (2.23 + 0.107i)5-s + 0.992i·7-s + (0.951 − 0.309i)8-s + (−1.22 − 1.87i)10-s + (2.58 − 1.87i)11-s + (−2.26 + 3.11i)13-s + (0.802 − 0.583i)14-s + (−0.809 − 0.587i)16-s + (2.15 − 0.701i)17-s + (1.45 + 4.46i)19-s + (−0.792 + 2.09i)20-s + (−3.03 − 0.986i)22-s + (3.29 + 4.53i)23-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (−0.154 + 0.475i)4-s + (0.998 + 0.0481i)5-s + 0.375i·7-s + (0.336 − 0.109i)8-s + (−0.387 − 0.591i)10-s + (0.778 − 0.565i)11-s + (−0.628 + 0.864i)13-s + (0.214 − 0.155i)14-s + (−0.202 − 0.146i)16-s + (0.523 − 0.170i)17-s + (0.332 + 1.02i)19-s + (−0.177 + 0.467i)20-s + (−0.647 − 0.210i)22-s + (0.686 + 0.944i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.959 + 0.280i$
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: \((2,\ 450,\ (\ :1/2),\ 0.959 + 0.280i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35472 - 0.194249i\)
\(L(\frac12)\) \(\approx\) \(1.35472 - 0.194249i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.107i)T \)
good7 \( 1 - 0.992iT - 7T^{2} \)
11 \( 1 + (-2.58 + 1.87i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (2.26 - 3.11i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.15 + 0.701i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.45 - 4.46i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.29 - 4.53i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.25 + 6.95i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.80 + 5.56i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.07 + 4.22i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.919 + 0.667i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.88iT - 43T^{2} \)
47 \( 1 + (6.88 + 2.23i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.00 - 1.30i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (5.55 + 4.03i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-8.95 + 6.50i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (12.4 - 4.04i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-0.675 + 2.07i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.67 - 2.30i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.44 - 10.6i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (9.87 - 3.20i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-1.56 + 1.13i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (15.7 + 5.11i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.14624887181861531241982788530, −9.730534597721132255946917330888, −9.654203296617683564673528010751, −8.600077116567563152628080586952, −7.47539373836415238152747797919, −6.31433291732364975248395974766, −5.41515707216501379564089482904, −3.99273654817212051911911875491, −2.66205565584239356347096931804, −1.45495347662863014008820239785, 1.23952174095430078414624424099, 2.88835853992272415529120495641, 4.67976882750519382571072903082, 5.47824610698716070745395522222, 6.70044432707983309057180959444, 7.22140253132082791758636888314, 8.558800261493193294142080831324, 9.279662825884443520055431887159, 10.17444096549345425172237213221, 10.70917065518403403123976552657

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