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Properties

Label 2-45-5.2-c2-0-3
Degree $2$
Conductor $45$
Sign $-0.640 + 0.767i$
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  + (−1.58 − 1.58i)2-s + 1.00i·4-s + (−1.58 − 4.74i)5-s + (−5 − 5i)7-s + (−4.74 + 4.74i)8-s + (−5 + 10.0i)10-s + 15.8·11-s + (10 − 10i)13-s + 15.8i·14-s + 19·16-s + (3.16 + 3.16i)17-s + 18i·19-s + (4.74 − 1.58i)20-s + (−25 − 25i)22-s + (3.16 − 3.16i)23-s + ⋯
L(s)  = 1  + (−0.790 − 0.790i)2-s + 0.250i·4-s + (−0.316 − 0.948i)5-s + (−0.714 − 0.714i)7-s + (−0.592 + 0.592i)8-s + (−0.5 + 1.00i)10-s + 1.43·11-s + (0.769 − 0.769i)13-s + 1.12i·14-s + 1.18·16-s + (0.186 + 0.186i)17-s + 0.947i·19-s + (0.237 − 0.0790i)20-s + (−1.13 − 1.13i)22-s + (0.137 − 0.137i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.640 + 0.767i$
Analytic conductor: \(1.22616\)
Root analytic conductor: \(1.10732\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1),\ -0.640 + 0.767i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.282755 - 0.604270i\)
\(L(\frac12)\) \(\approx\) \(0.282755 - 0.604270i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (1.58 + 4.74i)T \)
good2 \( 1 + (1.58 + 1.58i)T + 4iT^{2} \)
7 \( 1 + (5 + 5i)T + 49iT^{2} \)
11 \( 1 - 15.8T + 121T^{2} \)
13 \( 1 + (-10 + 10i)T - 169iT^{2} \)
17 \( 1 + (-3.16 - 3.16i)T + 289iT^{2} \)
19 \( 1 - 18iT - 361T^{2} \)
23 \( 1 + (-3.16 + 3.16i)T - 529iT^{2} \)
29 \( 1 + 47.4iT - 841T^{2} \)
31 \( 1 - 8T + 961T^{2} \)
37 \( 1 + (-10 - 10i)T + 1.36e3iT^{2} \)
41 \( 1 + 31.6T + 1.68e3T^{2} \)
43 \( 1 + (-10 + 10i)T - 1.84e3iT^{2} \)
47 \( 1 + (-41.1 - 41.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (25.2 - 25.2i)T - 2.80e3iT^{2} \)
59 \( 1 - 47.4iT - 3.48e3T^{2} \)
61 \( 1 + 58T + 3.72e3T^{2} \)
67 \( 1 + (-70 - 70i)T + 4.48e3iT^{2} \)
71 \( 1 - 63.2T + 5.04e3T^{2} \)
73 \( 1 + (-55 + 55i)T - 5.32e3iT^{2} \)
79 \( 1 - 12iT - 6.24e3T^{2} \)
83 \( 1 + (53.7 - 53.7i)T - 6.88e3iT^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + (5 + 5i)T + 9.40e3iT^{2} \)
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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

−15.35778838387373975982491318513, −13.86588117448618704821133706944, −12.54584940612755552450589198395, −11.55497479671012218091792319421, −10.22841955442148026022444801860, −9.260270254761943393254256602495, −8.107359053703639145676963085828, −6.05875941023193682378623855277, −3.82324094743719083889806152585, −1.02551862370993760166493414529, 3.46164702436307439002314720361, 6.34599495110069946504508343750, 7.01301774144573481201652635605, 8.715102600631252747470021527276, 9.531570061267914645515259405950, 11.25567763653390900346230388590, 12.38260105783380688891419048359, 14.08158237580846356865275245363, 15.22377634246544515319830255750, 16.04267945303262888143365657873

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