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Properties

Label 2-45-9.7-c1-0-1
Degree $2$
Conductor $45$
Sign $-0.391 - 0.920i$
Analytic cond. $0.359326$
Root an. cond. $0.599438$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.25 + 2.17i)2-s + (1.25 + 1.19i)3-s + (−2.16 − 3.74i)4-s + (−0.5 − 0.866i)5-s + (−4.17 + 1.23i)6-s + (0.257 − 0.445i)7-s + 5.83·8-s + (0.160 + 2.99i)9-s + 2.51·10-s + (1.66 − 2.87i)11-s + (1.74 − 7.27i)12-s + (0.660 + 1.14i)13-s + (0.646 + 1.11i)14-s + (0.403 − 1.68i)15-s + (−3.01 + 5.22i)16-s − 3.32·17-s + ⋯
L(s)  = 1  + (−0.888 + 1.53i)2-s + (0.725 + 0.687i)3-s + (−1.08 − 1.87i)4-s + (−0.223 − 0.387i)5-s + (−1.70 + 0.505i)6-s + (0.0971 − 0.168i)7-s + 2.06·8-s + (0.0534 + 0.998i)9-s + 0.795·10-s + (0.500 − 0.867i)11-s + (0.503 − 2.10i)12-s + (0.183 + 0.317i)13-s + (0.172 + 0.299i)14-s + (0.104 − 0.434i)15-s + (−0.753 + 1.30i)16-s − 0.805·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.391 - 0.920i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1/2),\ -0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.340195 + 0.514619i\)
\(L(\frac12)\) \(\approx\) \(0.340195 + 0.514619i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.25 - 1.19i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (1.25 - 2.17i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-0.257 + 0.445i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.66 + 2.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.660 - 1.14i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.32T + 17T^{2} \)
19 \( 1 + 1.32T + 19T^{2} \)
23 \( 1 + (2.06 + 3.57i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.693 + 1.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.36 + 7.56i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.292T + 37T^{2} \)
41 \( 1 + (-5.67 - 9.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.17 - 8.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.43 + 4.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.02T + 53T^{2} \)
59 \( 1 + (-2.51 - 4.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.67 - 6.36i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.72 + 8.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.99T + 71T^{2} \)
73 \( 1 - 6.05T + 73T^{2} \)
79 \( 1 + (-4.02 + 6.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.771 - 1.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (-6.12 + 10.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−16.37410673108844327852605382890, −15.30949872459198978891896732132, −14.45417159804209492825406834547, −13.43209947858917081083562770961, −11.03798828521562930290166860047, −9.576985795422981664728550952154, −8.716217899067564460578999369303, −7.82124523891298629099244885430, −6.18808179587790364166996601730, −4.44228803956075777510565458328, 2.04292494232949919794083334528, 3.68429497812206630325854139835, 7.12962302985200273146768698065, 8.476617784715382467892862713707, 9.444576773517806221267865412173, 10.74581902953280784113595073752, 11.98755644733206073316711613288, 12.75074445854338322993393155238, 14.02930369509783194636022879366, 15.41146440760171134088747680497

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