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Properties

Label 2-450-45.2-c1-0-5
Degree $2$
Conductor $450$
Sign $0.278 - 0.960i$
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  + (0.258 + 0.965i)2-s + (−1.62 + 0.599i)3-s + (−0.866 + 0.499i)4-s + (−1 − 1.41i)6-s + (4.40 − 1.18i)7-s + (−0.707 − 0.707i)8-s + (2.28 − 1.94i)9-s + (−0.550 − 0.317i)11-s + (1.10 − 1.33i)12-s + (3.34 + 0.896i)13-s + (2.28 + 3.94i)14-s + (0.500 − 0.866i)16-s + (−0.317 + 0.317i)17-s + (2.47 + 1.69i)18-s + 6.44i·19-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.938 + 0.346i)3-s + (−0.433 + 0.249i)4-s + (−0.408 − 0.577i)6-s + (1.66 − 0.446i)7-s + (−0.249 − 0.249i)8-s + (0.760 − 0.649i)9-s + (−0.165 − 0.0958i)11-s + (0.319 − 0.384i)12-s + (0.928 + 0.248i)13-s + (0.609 + 1.05i)14-s + (0.125 − 0.216i)16-s + (−0.0770 + 0.0770i)17-s + (0.582 + 0.400i)18-s + 1.47i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.278 - 0.960i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.278 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05607 + 0.793088i\)
\(L(\frac12)\) \(\approx\) \(1.05607 + 0.793088i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (1.62 - 0.599i)T \)
5 \( 1 \)
good7 \( 1 + (-4.40 + 1.18i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.550 + 0.317i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.34 - 0.896i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.317 - 0.317i)T - 17iT^{2} \)
19 \( 1 - 6.44iT - 19T^{2} \)
23 \( 1 + (-0.258 + 0.965i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.158 - 0.275i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.224 + 0.389i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3 - 3i)T + 37iT^{2} \)
41 \( 1 + (-6.39 + 3.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.896 - 3.34i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.32 - 8.69i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.78 + 3.78i)T + 53iT^{2} \)
59 \( 1 + (4.48 + 7.77i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.275 + 0.476i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.71 + 6.38i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 6.29iT - 71T^{2} \)
73 \( 1 + (-6.89 + 6.89i)T - 73iT^{2} \)
79 \( 1 + (2.12 + 1.22i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.26 - 1.41i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 8.02T + 89T^{2} \)
97 \( 1 + (-2.59 + 0.695i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.14410339422671742857607329228, −10.65489077642635586558703071637, −9.477188440132685772999814489977, −8.272068635404070655448173930867, −7.65045590647401785810565236041, −6.42014317926077110222500434002, −5.59714349726953349854864446512, −4.64811039037155288436586154338, −3.87718060518937104688325882736, −1.37523153071072145077868778629, 1.13513997621708988247559467526, 2.38147510110409993279539022678, 4.27056287314976509738925932514, 5.09613220576042151324586246553, 5.88391465346115219431326956751, 7.22702974032688983233424859040, 8.219280096124776323083175024795, 9.135223595404636406784507557362, 10.47800913263924880700264345631, 11.19452317308033456401532889539

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