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Properties

Label 2-450-45.23-c1-0-3
Degree $2$
Conductor $450$
Sign $0.148 - 0.988i$
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 + (−1.45 − 0.933i)3-s + (−0.866 − 0.499i)4-s + (1.27 − 1.16i)6-s + (−1.94 − 0.521i)7-s + (0.707 − 0.707i)8-s + (1.25 + 2.72i)9-s + (−1.70 + 0.984i)11-s + (0.796 + 1.53i)12-s + (3.92 − 1.05i)13-s + (1.00 − 1.74i)14-s + (0.500 + 0.866i)16-s + (2.35 + 2.35i)17-s + (−2.95 + 0.507i)18-s + 3.70i·19-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.842 − 0.539i)3-s + (−0.433 − 0.249i)4-s + (0.522 − 0.476i)6-s + (−0.736 − 0.197i)7-s + (0.249 − 0.249i)8-s + (0.418 + 0.908i)9-s + (−0.514 + 0.296i)11-s + (0.229 + 0.444i)12-s + (1.08 − 0.291i)13-s + (0.269 − 0.466i)14-s + (0.125 + 0.216i)16-s + (0.572 + 0.572i)17-s + (−0.696 + 0.119i)18-s + 0.850i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.148 - 0.988i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.148 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.569439 + 0.490254i\)
\(L(\frac12)\) \(\approx\) \(0.569439 + 0.490254i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.45 + 0.933i)T \)
5 \( 1 \)
good7 \( 1 + (1.94 + 0.521i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.70 - 0.984i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.92 + 1.05i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-2.35 - 2.35i)T + 17iT^{2} \)
19 \( 1 - 3.70iT - 19T^{2} \)
23 \( 1 + (-1.62 - 6.05i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.74 - 6.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.48 + 6.04i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.26 - 4.26i)T - 37iT^{2} \)
41 \( 1 + (-6.13 - 3.54i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.43 - 9.09i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.00 - 7.49i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-7.03 + 7.03i)T - 53iT^{2} \)
59 \( 1 + (1.34 - 2.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.37 + 7.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.19 + 8.18i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 5.68iT - 71T^{2} \)
73 \( 1 + (1.14 + 1.14i)T + 73iT^{2} \)
79 \( 1 + (-10.0 + 5.80i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.64 - 0.440i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 2.04T + 89T^{2} \)
97 \( 1 + (-9.71 - 2.60i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.20117926543740454837952397043, −10.35047502988442869595310287434, −9.613855392668118891140163661237, −8.224769817088563095370005385762, −7.60409359154952057992125737458, −6.47198260268689553589916319236, −5.93815355531285281755698555513, −4.88937101527935455600398216131, −3.43389022639425138851515493480, −1.29597710810221982868821090583, 0.64483198341795280939737154204, 2.78892952312634527623842436654, 3.91426795988206061725420491505, 5.03327470172079603568487018692, 6.07181280199666960599377084860, 7.01681492751838129100312760600, 8.563208580704205754012538468019, 9.231152502409297872606667844211, 10.31241625362582111504434044221, 10.70575988783941965644002967570

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