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Properties

Label 2-45-5.3-c8-0-16
Degree $2$
Conductor $45$
Sign $-0.940 + 0.339i$
Analytic cond. $18.3320$
Root an. cond. $4.28159$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (10.7 − 10.7i)2-s + 23.0i·4-s + (−489. − 388. i)5-s + (2.35e3 − 2.35e3i)7-s + (3.01e3 + 3.01e3i)8-s + (−9.47e3 + 1.09e3i)10-s − 1.19e4·11-s + (−3.71e4 − 3.71e4i)13-s − 5.08e4i·14-s + 5.91e4·16-s + (−7.70e4 + 7.70e4i)17-s − 1.72e5i·19-s + (8.93e3 − 1.12e4i)20-s + (−1.28e5 + 1.28e5i)22-s + (−3.03e5 − 3.03e5i)23-s + ⋯
L(s)  = 1  + (0.674 − 0.674i)2-s + 0.0899i·4-s + (−0.783 − 0.621i)5-s + (0.980 − 0.980i)7-s + (0.735 + 0.735i)8-s + (−0.947 + 0.109i)10-s − 0.813·11-s + (−1.30 − 1.30i)13-s − 1.32i·14-s + 0.902·16-s + (−0.922 + 0.922i)17-s − 1.32i·19-s + (0.0558 − 0.0704i)20-s + (−0.549 + 0.549i)22-s + (−1.08 − 1.08i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.339i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.940 + 0.339i$
Analytic conductor: \(18.3320\)
Root analytic conductor: \(4.28159\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :4),\ -0.940 + 0.339i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.272737 - 1.55757i\)
\(L(\frac12)\) \(\approx\) \(0.272737 - 1.55757i\)
\(L(5)\) 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 \( 1 + (489. + 388. i)T \)
good2 \( 1 + (-10.7 + 10.7i)T - 256iT^{2} \)
7 \( 1 + (-2.35e3 + 2.35e3i)T - 5.76e6iT^{2} \)
11 \( 1 + 1.19e4T + 2.14e8T^{2} \)
13 \( 1 + (3.71e4 + 3.71e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (7.70e4 - 7.70e4i)T - 6.97e9iT^{2} \)
19 \( 1 + 1.72e5iT - 1.69e10T^{2} \)
23 \( 1 + (3.03e5 + 3.03e5i)T + 7.83e10iT^{2} \)
29 \( 1 + 3.61e5iT - 5.00e11T^{2} \)
31 \( 1 - 4.32e5T + 8.52e11T^{2} \)
37 \( 1 + (-6.98e5 + 6.98e5i)T - 3.51e12iT^{2} \)
41 \( 1 - 2.69e6T + 7.98e12T^{2} \)
43 \( 1 + (-3.37e5 - 3.37e5i)T + 1.16e13iT^{2} \)
47 \( 1 + (1.97e6 - 1.97e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (-3.56e6 - 3.56e6i)T + 6.22e13iT^{2} \)
59 \( 1 + 9.30e6iT - 1.46e14T^{2} \)
61 \( 1 - 1.35e7T + 1.91e14T^{2} \)
67 \( 1 + (-2.69e5 + 2.69e5i)T - 4.06e14iT^{2} \)
71 \( 1 - 4.06e7T + 6.45e14T^{2} \)
73 \( 1 + (7.18e5 + 7.18e5i)T + 8.06e14iT^{2} \)
79 \( 1 + 2.64e7iT - 1.51e15T^{2} \)
83 \( 1 + (5.69e7 + 5.69e7i)T + 2.25e15iT^{2} \)
89 \( 1 - 1.47e7iT - 3.93e15T^{2} \)
97 \( 1 + (-5.88e7 + 5.88e7i)T - 7.83e15iT^{2} \)
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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

−13.21152230103914861112905916911, −12.53978663165426164793353798713, −11.29821867031419207088884056091, −10.46077579587285461914445537874, −8.235868001306621500372278701741, −7.58754292944545779031232338771, −4.93855593958676870460452407921, −4.21834682817949045267547433507, −2.45806249573937190883321393535, −0.45247876872694219299884560974, 2.18410666293520807781681221273, 4.32364132596574347908755009835, 5.42965070371940453331841914912, 6.96118650738644854757418949621, 8.019341692941785444301129428343, 9.802230290499541131043710316708, 11.32142940940559851117496368619, 12.20865352210460345827509728004, 13.94053272541586901847190908177, 14.65310967096289556945910476087

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