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Properties

Label 2-450-45.32-c1-0-14
Degree $2$
Conductor $450$
Sign $0.395 + 0.918i$
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  + (0.965 + 0.258i)2-s + (−0.448 − 1.67i)3-s + (0.866 + 0.499i)4-s − 1.73i·6-s + (0.448 − 1.67i)7-s + (0.707 + 0.707i)8-s + (−2.59 + 1.50i)9-s + (3 − 1.73i)11-s + (0.448 − 1.67i)12-s + (−0.896 − 3.34i)13-s + (0.866 − 1.50i)14-s + (0.500 + 0.866i)16-s + (4.24 − 4.24i)17-s + (−2.89 + 0.776i)18-s + 2i·19-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.258 − 0.965i)3-s + (0.433 + 0.249i)4-s − 0.707i·6-s + (0.169 − 0.632i)7-s + (0.249 + 0.249i)8-s + (−0.866 + 0.5i)9-s + (0.904 − 0.522i)11-s + (0.129 − 0.482i)12-s + (−0.248 − 0.928i)13-s + (0.231 − 0.400i)14-s + (0.125 + 0.216i)16-s + (1.02 − 1.02i)17-s + (−0.683 + 0.183i)18-s + 0.458i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.395 + 0.918i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.395 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60888 - 1.05920i\)
\(L(\frac12)\) \(\approx\) \(1.60888 - 1.05920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (0.448 + 1.67i)T \)
5 \( 1 \)
good7 \( 1 + (-0.448 + 1.67i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.896 + 3.34i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-4.24 + 4.24i)T - 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (2.89 - 0.776i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (4.33 + 7.5i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.44 + 2.44i)T + 37iT^{2} \)
41 \( 1 + (-7.5 - 4.33i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-10.0 - 2.68i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.89 - 0.776i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + (3.46 - 6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.7 - 3.13i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + (9.79 - 9.79i)T - 73iT^{2} \)
79 \( 1 + (-3.46 + 2i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.776 - 2.89i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 1.73T + 89T^{2} \)
97 \( 1 + (1.79 - 6.69i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.21563924906451613195042657303, −10.27203774563272819612747461493, −8.951127482599211141572338060034, −7.64121379657022880839751494077, −7.39280962323551852859261472518, −6.06402558557900415091779660103, −5.47968446113705484044200965911, −4.03021855187867723714143243957, −2.81023808622738711124849137482, −1.11468177984594268589187947887, 2.02927012514411924923198867677, 3.61557378293176581882581954876, 4.34886380241188043163010848816, 5.47294209927216360402295529218, 6.20590154404856826699275541365, 7.44898817554324641891817593719, 8.923074055287675558711040889238, 9.466630701224255095834461774848, 10.51567391100950217389696577456, 11.33163974372868500873273960875

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