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Properties

Label 2-45-5.4-c3-0-5
Degree $2$
Conductor $45$
Sign $-0.993 - 0.116i$
Analytic cond. $2.65508$
Root an. cond. $1.62944$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.70i·2-s − 14.1·4-s + (−11.1 − 1.29i)5-s − 16.2i·7-s + 28.7i·8-s + (−6.10 + 52.2i)10-s + 40.2·11-s − 19.7i·13-s − 76.2·14-s + 22.1·16-s − 83.0i·17-s + 48.8·19-s + (156. + 18.3i)20-s − 189. i·22-s + 1.61i·23-s + ⋯
L(s)  = 1  − 1.66i·2-s − 1.76·4-s + (−0.993 − 0.116i)5-s − 0.875i·7-s + 1.26i·8-s + (−0.193 + 1.65i)10-s + 1.10·11-s − 0.422i·13-s − 1.45·14-s + 0.345·16-s − 1.18i·17-s + 0.589·19-s + (1.75 + 0.204i)20-s − 1.83i·22-s + 0.0146i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.993 - 0.116i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ -0.993 - 0.116i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0563717 + 0.967505i\)
\(L(\frac12)\) \(\approx\) \(0.0563717 + 0.967505i\)
\(L(\frac{5}{2})\) 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 \( 1 + (11.1 + 1.29i)T \)
good2 \( 1 + 4.70iT - 8T^{2} \)
7 \( 1 + 16.2iT - 343T^{2} \)
11 \( 1 - 40.2T + 1.33e3T^{2} \)
13 \( 1 + 19.7iT - 2.19e3T^{2} \)
17 \( 1 + 83.0iT - 4.91e3T^{2} \)
19 \( 1 - 48.8T + 6.85e3T^{2} \)
23 \( 1 - 1.61iT - 1.21e4T^{2} \)
29 \( 1 + 24.5T + 2.43e4T^{2} \)
31 \( 1 + 12.4T + 2.97e4T^{2} \)
37 \( 1 - 325. iT - 5.06e4T^{2} \)
41 \( 1 - 242.T + 6.89e4T^{2} \)
43 \( 1 + 367. iT - 7.95e4T^{2} \)
47 \( 1 - 204. iT - 1.03e5T^{2} \)
53 \( 1 + 61.5iT - 1.48e5T^{2} \)
59 \( 1 + 112.T + 2.05e5T^{2} \)
61 \( 1 - 477.T + 2.26e5T^{2} \)
67 \( 1 - 558. iT - 3.00e5T^{2} \)
71 \( 1 + 558.T + 3.57e5T^{2} \)
73 \( 1 + 1.01e3iT - 3.89e5T^{2} \)
79 \( 1 + 1.15e3T + 4.93e5T^{2} \)
83 \( 1 + 1.15e3iT - 5.71e5T^{2} \)
89 \( 1 - 96.9T + 7.04e5T^{2} \)
97 \( 1 + 1.15e3iT - 9.12e5T^{2} \)
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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

−14.33939320812975226713791702279, −13.26274819213481221374791877446, −11.99682055343390217915493443166, −11.38954482900435083956603886387, −10.19387517031200527746003192328, −8.961745637624656302324675940371, −7.28957686173416157598035105732, −4.50223222199671476751873694112, −3.33139725979925337978033432666, −0.845502419906100721065398154315, 4.14107489208732795074021262122, 5.83349246435197788964917444805, 7.04765906841444647643904145975, 8.281653265664983650020886053974, 9.222983683466609824833945714863, 11.39087526075917559707042000764, 12.60655274570633196421930544039, 14.28783438670775530346046355818, 14.95474164046118177793178382211, 15.86558513285292312656605313904

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