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Properties

Label 2-45-9.5-c2-0-3
Degree $2$
Conductor $45$
Sign $0.785 - 0.618i$
Analytic cond. $1.22616$
Root an. cond. $1.10732$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.23 + 1.86i)2-s + (2.62 − 1.45i)3-s + (4.98 − 8.62i)4-s + (1.93 + 1.11i)5-s + (−5.75 + 9.61i)6-s + (3.63 + 6.28i)7-s + 22.2i·8-s + (4.74 − 7.65i)9-s − 8.35·10-s + (6.50 − 3.75i)11-s + (0.466 − 29.8i)12-s + (−1.46 + 2.54i)13-s + (−23.4 − 13.5i)14-s + (6.70 + 0.104i)15-s + (−21.6 − 37.5i)16-s − 1.90i·17-s + ⋯
L(s)  = 1  + (−1.61 + 0.934i)2-s + (0.873 − 0.486i)3-s + (1.24 − 2.15i)4-s + (0.387 + 0.223i)5-s + (−0.959 + 1.60i)6-s + (0.518 + 0.898i)7-s + 2.78i·8-s + (0.526 − 0.850i)9-s − 0.835·10-s + (0.591 − 0.341i)11-s + (0.0388 − 2.49i)12-s + (−0.112 + 0.195i)13-s + (−1.67 − 0.968i)14-s + (0.447 + 0.00697i)15-s + (−1.35 − 2.34i)16-s − 0.111i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 - 0.618i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.785 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.785 - 0.618i$
Analytic conductor: \(1.22616\)
Root analytic conductor: \(1.10732\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1),\ 0.785 - 0.618i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.726655 + 0.251710i\)
\(L(\frac12)\) \(\approx\) \(0.726655 + 0.251710i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.62 + 1.45i)T \)
5 \( 1 + (-1.93 - 1.11i)T \)
good2 \( 1 + (3.23 - 1.86i)T + (2 - 3.46i)T^{2} \)
7 \( 1 + (-3.63 - 6.28i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-6.50 + 3.75i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (1.46 - 2.54i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 1.90iT - 289T^{2} \)
19 \( 1 + 7.38T + 361T^{2} \)
23 \( 1 + (30.6 + 17.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (14.2 - 8.19i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (13.3 - 23.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 44.6T + 1.36e3T^{2} \)
41 \( 1 + (-14.8 - 8.58i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (20.5 + 35.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-36.4 + 21.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 100. iT - 2.80e3T^{2} \)
59 \( 1 + (4.12 + 2.37i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-38.4 - 66.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-41.9 + 72.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 23.1iT - 5.04e3T^{2} \)
73 \( 1 + 103.T + 5.32e3T^{2} \)
79 \( 1 + (40.2 + 69.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (17.1 - 9.87i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 29.0iT - 7.92e3T^{2} \)
97 \( 1 + (-47.7 - 82.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−15.78032398692467781449558014954, −14.79785966311174537073439214832, −14.06288593718942109971679352950, −12.01559891759074761692473089096, −10.39378759117743544487135008802, −9.080544222047686417079177786104, −8.500339635203907521238234797535, −7.18155665066771438957135480784, −5.97832617624226660215638933067, −1.95151857852107709319293107287, 1.86301095099191923464900489537, 3.88898068558197555510055064033, 7.38212754863796914772020439145, 8.386197818689748369081969651580, 9.563082878047494108998259130809, 10.28261998812810565901675838975, 11.45020794045825996559175168266, 12.96689111490769405272335184323, 14.32072804572536857563112854193, 15.88872802762570748974500558160

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