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Properties

Label 2-45-45.14-c2-0-8
Degree $2$
Conductor $45$
Sign $-0.0268 + 0.999i$
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  + (−0.264 − 0.457i)2-s + (−2.94 − 0.546i)3-s + (1.86 − 3.22i)4-s + (−0.819 − 4.93i)5-s + (0.529 + 1.49i)6-s + (2.39 − 1.38i)7-s − 4.08·8-s + (8.40 + 3.22i)9-s + (−2.04 + 1.67i)10-s + (−7.99 + 4.61i)11-s + (−7.24 + 8.48i)12-s + (11.7 + 6.79i)13-s + (−1.26 − 0.731i)14-s + (−0.275 + 14.9i)15-s + (−6.36 − 11.0i)16-s + 12.2·17-s + ⋯
L(s)  = 1  + (−0.132 − 0.228i)2-s + (−0.983 − 0.182i)3-s + (0.465 − 0.805i)4-s + (−0.163 − 0.986i)5-s + (0.0883 + 0.249i)6-s + (0.342 − 0.197i)7-s − 0.510·8-s + (0.933 + 0.357i)9-s + (−0.204 + 0.167i)10-s + (−0.726 + 0.419i)11-s + (−0.603 + 0.707i)12-s + (0.905 + 0.522i)13-s + (−0.0904 − 0.0522i)14-s + (−0.0183 + 0.999i)15-s + (−0.397 − 0.688i)16-s + 0.718·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0268 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0268 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.0268 + 0.999i$
Analytic conductor: \(1.22616\)
Root analytic conductor: \(1.10732\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1),\ -0.0268 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.590075 - 0.606142i\)
\(L(\frac12)\) \(\approx\) \(0.590075 - 0.606142i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.94 + 0.546i)T \)
5 \( 1 + (0.819 + 4.93i)T \)
good2 \( 1 + (0.264 + 0.457i)T + (-2 + 3.46i)T^{2} \)
7 \( 1 + (-2.39 + 1.38i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (7.99 - 4.61i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-11.7 - 6.79i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 12.2T + 289T^{2} \)
19 \( 1 - 20.2T + 361T^{2} \)
23 \( 1 + (-1.18 + 2.05i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-30.2 + 17.4i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (14.7 - 25.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 64.3iT - 1.36e3T^{2} \)
41 \( 1 + (-34.5 - 19.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (58.5 - 33.7i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-46.6 - 80.8i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 9.82T + 2.80e3T^{2} \)
59 \( 1 + (50.6 + 29.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-7.75 - 13.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-13.4 - 7.78i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 53.1iT - 5.04e3T^{2} \)
73 \( 1 - 23.6iT - 5.32e3T^{2} \)
79 \( 1 + (17.2 + 29.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (37.6 + 65.2i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 29.1iT - 7.92e3T^{2} \)
97 \( 1 + (-54.0 + 31.1i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−15.81539831347632139107942965936, −14.09149608822447752017075200964, −12.69353225877421921418533363155, −11.68109525603225862474948363979, −10.70617838885786134720666726326, −9.483223962644938215894795746335, −7.62662811044669404273722525300, −5.97230338397255632390341613318, −4.82334484254527890910471385314, −1.23739139284384958614677443159, 3.35060139408119728236036008176, 5.65137793829069887936179797031, 6.97583257645533694488713138437, 8.159117300027897720386448816613, 10.24415717246914362517933182415, 11.25622266843112344467808824360, 12.06219080731680356012510172686, 13.49712467464394461092781349153, 15.22534996302718978577428633576, 15.91307288501522323118606985560

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