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Properties

Label 2-45-9.5-c2-0-0
Degree $2$
Conductor $45$
Sign $-0.470 + 0.882i$
Analytic cond. $1.22616$
Root an. cond. $1.10732$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.83 + 1.63i)2-s + (−2.26 + 1.96i)3-s + (3.34 − 5.79i)4-s + (−1.93 − 1.11i)5-s + (3.19 − 9.27i)6-s + (−3.16 − 5.48i)7-s + 8.82i·8-s + (1.25 − 8.91i)9-s + 7.31·10-s + (−12.8 + 7.43i)11-s + (3.82 + 19.7i)12-s + (−7.73 + 13.4i)13-s + (17.9 + 10.3i)14-s + (6.58 − 1.27i)15-s + (−1.03 − 1.78i)16-s − 20.0i·17-s + ⋯
L(s)  = 1  + (−1.41 + 0.817i)2-s + (−0.754 + 0.655i)3-s + (0.837 − 1.44i)4-s + (−0.387 − 0.223i)5-s + (0.532 − 1.54i)6-s + (−0.452 − 0.784i)7-s + 1.10i·8-s + (0.139 − 0.990i)9-s + 0.731·10-s + (−1.17 + 0.676i)11-s + (0.318 + 1.64i)12-s + (−0.595 + 1.03i)13-s + (1.28 + 0.740i)14-s + (0.439 − 0.0851i)15-s + (−0.0644 − 0.111i)16-s − 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.470 + 0.882i$
Analytic conductor: \(1.22616\)
Root analytic conductor: \(1.10732\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1),\ -0.470 + 0.882i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00192886 - 0.00321288i\)
\(L(\frac12)\) \(\approx\) \(0.00192886 - 0.00321288i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.26 - 1.96i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
good2 \( 1 + (2.83 - 1.63i)T + (2 - 3.46i)T^{2} \)
7 \( 1 + (3.16 + 5.48i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (12.8 - 7.43i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (7.73 - 13.4i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 20.0iT - 289T^{2} \)
19 \( 1 + 25.1T + 361T^{2} \)
23 \( 1 + (-1.40 - 0.812i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (1.07 - 0.619i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (6.69 - 11.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 3.89T + 1.36e3T^{2} \)
41 \( 1 + (-50.3 - 29.0i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-13.6 - 23.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (54.2 - 31.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 18.2iT - 2.80e3T^{2} \)
59 \( 1 + (25.1 + 14.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (55.5 + 96.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-8.56 + 14.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 52.7iT - 5.04e3T^{2} \)
73 \( 1 + 71.0T + 5.32e3T^{2} \)
79 \( 1 + (30.2 + 52.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-70.2 + 40.5i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 6.34iT - 7.92e3T^{2} \)
97 \( 1 + (-7.84 - 13.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−16.40349693284001573753080860910, −15.86992949374497498658754486179, −14.72009898028063444658214020801, −12.73217810992993203728258960116, −11.13117792833209590273210186942, −10.09329064270417198066798835470, −9.250149769367525508467371578995, −7.59989074508671273476110968524, −6.57364525861825701427017018362, −4.66642668716263303468899962278, 0.00638518581264519111683174706, 2.54091480709835274913946479616, 5.82805635529211301383158690840, 7.64351372224443822881360280522, 8.560807667453721391636079366946, 10.34600214115953769732515223235, 10.93324655206348798050077254829, 12.27272532715843843828355054326, 12.94840871316465846444881856816, 15.23098869234519845614859838109

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