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Properties

Label 2-45-5.3-c4-0-1
Degree $2$
Conductor $45$
Sign $-0.988 + 0.152i$
Analytic cond. $4.65164$
Root an. cond. $2.15676$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.30 + 3.30i)2-s − 5.84i·4-s + (16.2 + 19.0i)5-s + (−33.1 + 33.1i)7-s + (−33.5 − 33.5i)8-s + (−116. − 9.14i)10-s − 55.3·11-s + (−161. − 161. i)13-s − 219. i·14-s + 315.·16-s + (−278. + 278. i)17-s + 179. i·19-s + (111. − 94.9i)20-s + (182. − 182. i)22-s + (398. + 398. i)23-s + ⋯
L(s)  = 1  + (−0.826 + 0.826i)2-s − 0.365i·4-s + (0.649 + 0.760i)5-s + (−0.676 + 0.676i)7-s + (−0.524 − 0.524i)8-s + (−1.16 − 0.0914i)10-s − 0.457·11-s + (−0.958 − 0.958i)13-s − 1.11i·14-s + 1.23·16-s + (−0.965 + 0.965i)17-s + 0.498i·19-s + (0.277 − 0.237i)20-s + (0.377 − 0.377i)22-s + (0.752 + 0.752i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.152i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.988 + 0.152i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ -0.988 + 0.152i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0463106 - 0.602205i\)
\(L(\frac12)\) \(\approx\) \(0.0463106 - 0.602205i\)
\(L(3)\) 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 \( 1 + (-16.2 - 19.0i)T \)
good2 \( 1 + (3.30 - 3.30i)T - 16iT^{2} \)
7 \( 1 + (33.1 - 33.1i)T - 2.40e3iT^{2} \)
11 \( 1 + 55.3T + 1.46e4T^{2} \)
13 \( 1 + (161. + 161. i)T + 2.85e4iT^{2} \)
17 \( 1 + (278. - 278. i)T - 8.35e4iT^{2} \)
19 \( 1 - 179. iT - 1.30e5T^{2} \)
23 \( 1 + (-398. - 398. i)T + 2.79e5iT^{2} \)
29 \( 1 + 547. iT - 7.07e5T^{2} \)
31 \( 1 - 1.53e3T + 9.23e5T^{2} \)
37 \( 1 + (1.66e3 - 1.66e3i)T - 1.87e6iT^{2} \)
41 \( 1 - 307.T + 2.82e6T^{2} \)
43 \( 1 + (-104. - 104. i)T + 3.41e6iT^{2} \)
47 \( 1 + (-346. + 346. i)T - 4.87e6iT^{2} \)
53 \( 1 + (-2.02e3 - 2.02e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 2.85e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.05e3T + 1.38e7T^{2} \)
67 \( 1 + (-3.75e3 + 3.75e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 1.42e3T + 2.54e7T^{2} \)
73 \( 1 + (-813. - 813. i)T + 2.83e7iT^{2} \)
79 \( 1 - 4.85e3iT - 3.89e7T^{2} \)
83 \( 1 + (-1.31e3 - 1.31e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 5.18e3iT - 6.27e7T^{2} \)
97 \( 1 + (-3.49e3 + 3.49e3i)T - 8.85e7iT^{2} \)
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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

−15.45988459771227120194570974637, −15.18501629118141527921193115294, −13.42268625242509900897155326720, −12.29233476518726498212066759095, −10.41566496806522188445027314337, −9.527880542174947504679549664908, −8.192313744403619327726217753507, −6.85362623242398426782394034845, −5.77510231760737933980729171999, −2.87903197388709268502414848795, 0.47522990852314533177845574414, 2.39722206624890765327319997718, 4.90617270341511289543356115699, 6.83817751915495093945962965409, 8.801821179682504293884687206692, 9.610437723833068878097457081457, 10.59294589746030939728602823723, 11.94626342609693382728993156316, 13.13507437807083544442801108711, 14.24140805495024294404280292623

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