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Properties

Label 2-45-45.38-c1-0-2
Degree $2$
Conductor $45$
Sign $0.985 + 0.170i$
Analytic cond. $0.359326$
Root an. cond. $0.599438$
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  + (0.186 − 0.0499i)2-s + (1.53 − 0.806i)3-s + (−1.69 + 0.981i)4-s + (−2.04 − 0.893i)5-s + (0.245 − 0.226i)6-s + (0.632 + 2.35i)7-s + (−0.540 + 0.540i)8-s + (1.69 − 2.47i)9-s + (−0.426 − 0.0641i)10-s + (−2.14 − 1.23i)11-s + (−1.81 + 2.87i)12-s + (−0.422 + 1.57i)13-s + (0.235 + 0.407i)14-s + (−3.86 + 0.282i)15-s + (1.88 − 3.27i)16-s + (0.403 + 0.403i)17-s + ⋯
L(s)  = 1  + (0.131 − 0.0352i)2-s + (0.885 − 0.465i)3-s + (−0.849 + 0.490i)4-s + (−0.916 − 0.399i)5-s + (0.100 − 0.0925i)6-s + (0.238 + 0.891i)7-s + (−0.191 + 0.191i)8-s + (0.566 − 0.823i)9-s + (−0.134 − 0.0202i)10-s + (−0.646 − 0.373i)11-s + (−0.523 + 0.829i)12-s + (−0.117 + 0.436i)13-s + (0.0629 + 0.108i)14-s + (−0.997 + 0.0728i)15-s + (0.472 − 0.818i)16-s + (0.0979 + 0.0979i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.985 + 0.170i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1/2),\ 0.985 + 0.170i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.843395 - 0.0722655i\)
\(L(\frac12)\) \(\approx\) \(0.843395 - 0.0722655i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.53 + 0.806i)T \)
5 \( 1 + (2.04 + 0.893i)T \)
good2 \( 1 + (-0.186 + 0.0499i)T + (1.73 - i)T^{2} \)
7 \( 1 + (-0.632 - 2.35i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (2.14 + 1.23i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.422 - 1.57i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.403 - 0.403i)T + 17iT^{2} \)
19 \( 1 + 4.28iT - 19T^{2} \)
23 \( 1 + (-6.82 - 1.82i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.20 - 5.55i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.97 + 3.41i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.171 - 0.171i)T - 37iT^{2} \)
41 \( 1 + (6.52 - 3.76i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.95 - 1.32i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.91 + 0.780i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.12 - 6.12i)T - 53iT^{2} \)
59 \( 1 + (2.27 + 3.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.235 - 0.408i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.65 - 0.443i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.50iT - 71T^{2} \)
73 \( 1 + (-6.88 - 6.88i)T + 73iT^{2} \)
79 \( 1 + (6.50 + 3.75i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.85 + 10.6i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 2.90T + 89T^{2} \)
97 \( 1 + (0.379 + 1.41i)T + (-84.0 + 48.5i)T^{2} \)
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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

−15.57054408431111275645791441484, −14.71341217093815769545143736435, −13.34866655699360315306199241637, −12.63769672860124598186814695771, −11.49596861863144305441438697106, −9.179237388731723501579144344548, −8.544340175786030015721421855936, −7.40425802778464981417182416787, −4.92754752185020384520696212780, −3.21939218442922471925428814424, 3.61896249127040887007475587932, 4.86688862183632899891884717176, 7.39477596190944703343819281460, 8.456429090018229660350434918857, 9.976005614845232100625842974184, 10.78940850557615796307973086992, 12.75974420927033405520912497223, 13.85471825835248330716911031787, 14.78597163194048070104720306109, 15.48759207724960288791790452513

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