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Properties

Label 2-45-5.2-c4-0-7
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: $\chi_{45} (37, \cdot )$
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

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

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