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Properties

Label 2-450-45.38-c1-0-10
Degree $2$
Conductor $450$
Sign $0.999 - 0.0253i$
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 + (1.73 + 0.0795i)3-s + (0.866 − 0.499i)4-s + (1.69 − 0.370i)6-s + (1.00 + 3.75i)7-s + (0.707 − 0.707i)8-s + (2.98 + 0.275i)9-s + (−3.44 − 1.98i)11-s + (1.53 − 0.796i)12-s + (0.256 − 0.956i)13-s + (1.94 + 3.36i)14-s + (0.500 − 0.866i)16-s + (−0.120 − 0.120i)17-s + (2.95 − 0.507i)18-s + 1.88i·19-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.998 + 0.0459i)3-s + (0.433 − 0.249i)4-s + (0.690 − 0.151i)6-s + (0.380 + 1.41i)7-s + (0.249 − 0.249i)8-s + (0.995 + 0.0917i)9-s + (−1.03 − 0.599i)11-s + (0.444 − 0.229i)12-s + (0.0710 − 0.265i)13-s + (0.519 + 0.899i)14-s + (0.125 − 0.216i)16-s + (−0.0291 − 0.0291i)17-s + (0.696 − 0.119i)18-s + 0.432i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.999 - 0.0253i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.999 - 0.0253i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.81069 + 0.0356431i\)
\(L(\frac12)\) \(\approx\) \(2.81069 + 0.0356431i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 0.0795i)T \)
5 \( 1 \)
good7 \( 1 + (-1.00 - 3.75i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (3.44 + 1.98i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.256 + 0.956i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.120 + 0.120i)T + 17iT^{2} \)
19 \( 1 - 1.88iT - 19T^{2} \)
23 \( 1 + (5.08 + 1.36i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.15 + 3.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.70 + 8.14i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.26 - 3.26i)T - 37iT^{2} \)
41 \( 1 + (-7.15 + 4.13i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.99 + 0.533i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (3.34 - 0.897i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.66 - 3.66i)T - 53iT^{2} \)
59 \( 1 + (2.72 + 4.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.35 - 7.54i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.86 + 2.10i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 6.94iT - 71T^{2} \)
73 \( 1 + (-8.27 - 8.27i)T + 73iT^{2} \)
79 \( 1 + (-11.7 - 6.78i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.81 + 6.75i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 4.87T + 89T^{2} \)
97 \( 1 + (-0.387 - 1.44i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.18250488377008344716381277466, −10.20622341481066197839564556078, −9.250975018665511049250387944326, −8.268628375639665296961320768119, −7.70700581046679091277860479513, −6.12389971451223067862978701101, −5.35927531419215710907940397629, −4.12141558846862068106929354053, −2.86160584751931882823924976852, −2.10554094528449617746393859394, 1.77811533210071908745799553897, 3.19116687569600712258214969082, 4.21461274108919473055688776336, 5.02412333596338850818487381220, 6.67192621542424833258110445394, 7.47812292852851102484831335938, 8.012671355879534159213364983038, 9.263696326941509032570191546889, 10.36691340828938176630072954525, 10.87377363440645241295230470887

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