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Properties

Label 2-450-25.11-c1-0-4
Degree $2$
Conductor $450$
Sign $0.834 - 0.550i$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (2.05 + 0.870i)5-s + 2.92·7-s + (0.309 + 0.951i)8-s + (−2.17 + 0.506i)10-s + (0.154 − 0.111i)11-s + (−0.250 − 0.182i)13-s + (−2.36 + 1.72i)14-s + (−0.809 − 0.587i)16-s + (−1.86 − 5.75i)17-s + (1 + 3.07i)19-s + (1.46 − 1.69i)20-s + (−0.0588 + 0.181i)22-s + (3.61 − 2.62i)23-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (0.921 + 0.389i)5-s + 1.10·7-s + (0.109 + 0.336i)8-s + (−0.688 + 0.160i)10-s + (0.0464 − 0.0337i)11-s + (−0.0695 − 0.0505i)13-s + (−0.633 + 0.459i)14-s + (−0.202 − 0.146i)16-s + (−0.453 − 1.39i)17-s + (0.229 + 0.706i)19-s + (0.327 − 0.377i)20-s + (−0.0125 + 0.0385i)22-s + (0.754 − 0.548i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.834 - 0.550i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.834 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32444 + 0.397363i\)
\(L(\frac12)\) \(\approx\) \(1.32444 + 0.397363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 \)
5 \( 1 + (-2.05 - 0.870i)T \)
good7 \( 1 - 2.92T + 7T^{2} \)
11 \( 1 + (-0.154 + 0.111i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.250 + 0.182i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.86 + 5.75i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1 - 3.07i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.61 + 2.62i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.10 - 6.47i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.522 + 1.60i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-7.89 - 5.73i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.10 - 2.98i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + (-3.31 + 10.1i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.08 - 9.48i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.00 - 4.36i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.48 + 3.25i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (4.66 + 14.3i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (2.11 - 6.51i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (9.41 - 6.84i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.485 + 1.49i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.02 + 12.3i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (7.51 - 5.45i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.537 + 1.65i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.05273923927741388878495839851, −10.17846431800887518888056343646, −9.339937310497434180790089361804, −8.513762466326290247363540168378, −7.46960605592557968562568977281, −6.67392694372474757976502864720, −5.53151776790557560448483274978, −4.73686738502403045187461606986, −2.78126350117539871836893432760, −1.45794438164964406362638616432, 1.36663779145951411689265621813, 2.41520654990894514142765577315, 4.15412399529327536957740846635, 5.23853910172501428320653393817, 6.33169712876556864536452378310, 7.57988264496352711125596234585, 8.476225440776902305736785862162, 9.211709128163722506437743314736, 10.08284154401052334308254107998, 11.02198013853441499813900892584

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