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Properties

Label 2-45-9.4-c3-0-1
Degree $2$
Conductor $45$
Sign $-0.917 - 0.396i$
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  + (2.65 + 4.60i)2-s + (−5.19 − 0.153i)3-s + (−10.1 + 17.5i)4-s + (2.5 − 4.33i)5-s + (−13.0 − 24.3i)6-s + (6.71 + 11.6i)7-s − 65.1·8-s + (26.9 + 1.59i)9-s + 26.5·10-s + (23.4 + 40.6i)11-s + (55.2 − 89.5i)12-s + (18.0 − 31.2i)13-s + (−35.7 + 61.8i)14-s + (−13.6 + 22.1i)15-s + (−92.1 − 159. i)16-s − 54.6·17-s + ⋯
L(s)  = 1  + (0.939 + 1.62i)2-s + (−0.999 − 0.0295i)3-s + (−1.26 + 2.19i)4-s + (0.223 − 0.387i)5-s + (−0.891 − 1.65i)6-s + (0.362 + 0.628i)7-s − 2.87·8-s + (0.998 + 0.0589i)9-s + 0.840·10-s + (0.643 + 1.11i)11-s + (1.33 − 2.15i)12-s + (0.385 − 0.667i)13-s + (−0.681 + 1.18i)14-s + (−0.234 + 0.380i)15-s + (−1.43 − 2.49i)16-s − 0.779·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.396i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.917 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.917 - 0.396i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ -0.917 - 0.396i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.309022 + 1.49337i\)
\(L(\frac12)\) \(\approx\) \(0.309022 + 1.49337i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.19 + 0.153i)T \)
5 \( 1 + (-2.5 + 4.33i)T \)
good2 \( 1 + (-2.65 - 4.60i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-6.71 - 11.6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-23.4 - 40.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-18.0 + 31.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 54.6T + 4.91e3T^{2} \)
19 \( 1 - 111.T + 6.85e3T^{2} \)
23 \( 1 + (-17.9 + 31.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (29.0 + 50.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-147. + 256. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 53.0T + 5.06e4T^{2} \)
41 \( 1 + (64.1 - 111. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-82.0 - 142. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (43.9 + 76.0i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 479.T + 1.48e5T^{2} \)
59 \( 1 + (317. - 550. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (24.0 + 41.5i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-14.4 + 25.0i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 576.T + 3.57e5T^{2} \)
73 \( 1 - 835.T + 3.89e5T^{2} \)
79 \( 1 + (-101. - 176. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (232. + 402. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 993.T + 7.04e5T^{2} \)
97 \( 1 + (440. + 763. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.67083106731892712548110947257, −15.06746573884330900338676853022, −13.60695329385530151164106633994, −12.62069908224304129413058880819, −11.70001814833352046534302903479, −9.442783227206022293013836091364, −7.85736625132148694176426423102, −6.55854191082922439452323150766, −5.46780537500053338123778032373, −4.43768604408612956409558559039, 1.22881869830455183534048733581, 3.63562525427623859740524397664, 5.05609208531751942197657172437, 6.48275467843947711513557146905, 9.380429503598857630621989420344, 10.73036028606387546550132738292, 11.24650692637445466138219134879, 12.18924393107212629598743982355, 13.62977116688768299540552849609, 14.11572585916576784331177178072

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