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Properties

Label 2-450-45.34-c1-0-8
Degree $2$
Conductor $450$
Sign $0.974 - 0.225i$
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  + (−0.866 − 0.5i)2-s + (1.65 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−1.18 − 1.26i)6-s + (2.92 + 1.68i)7-s − 0.999i·8-s + (2.5 + 1.65i)9-s + (2.18 − 3.78i)11-s + (0.396 + 1.68i)12-s + (−5.84 + 3.37i)13-s + (−1.68 − 2.92i)14-s + (−0.5 + 0.866i)16-s + 1.62i·17-s + (−1.33 − 2.68i)18-s + 2.37·19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.957 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.484 − 0.515i)6-s + (1.10 + 0.637i)7-s − 0.353i·8-s + (0.833 + 0.552i)9-s + (0.659 − 1.14i)11-s + (0.114 + 0.486i)12-s + (−1.61 + 0.935i)13-s + (−0.450 − 0.780i)14-s + (−0.125 + 0.216i)16-s + 0.394i·17-s + (−0.314 − 0.633i)18-s + 0.544·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.974 - 0.225i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.974 - 0.225i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58702 + 0.180927i\)
\(L(\frac12)\) \(\approx\) \(1.58702 + 0.180927i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-1.65 - 0.5i)T \)
5 \( 1 \)
good7 \( 1 + (-2.92 - 1.68i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.18 + 3.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.84 - 3.37i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.62iT - 17T^{2} \)
19 \( 1 - 2.37T + 19T^{2} \)
23 \( 1 + (-1.18 + 0.686i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.686 + 1.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.37 + 4.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.87 - 2.81i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.38 + 3.68i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 + (-2.18 - 3.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.05 + 7.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.06 + 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 3.11iT - 73T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.38 + 3.68i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 16.1T + 89T^{2} \)
97 \( 1 + (7.25 + 4.18i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.23416458043164366418841739534, −9.941623039171365335622136080780, −9.305512552014793603567001731701, −8.484765056679645390181052671906, −7.86606615159942126078972690275, −6.78042127285470718517563684591, −5.19339287001020303574397500233, −4.09410664864469971567880659271, −2.76650226012741293524039636937, −1.71441698804696823212687616108, 1.35234229984800100174665721324, 2.61955038720474418110173583704, 4.27916606057921145796717574631, 5.25185579079234113610294163919, 7.13118048476550920083093696592, 7.32115950431332777424645750482, 8.177813845358381849996163713626, 9.280126415950903333324587450669, 9.882704242944891013657108753909, 10.80001003492148309241679279316

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