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Properties

Label 2-45-9.4-c1-0-2
Degree $2$
Conductor $45$
Sign $0.173 + 0.984i$
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  + (−0.5 − 0.866i)2-s + (−1.5 − 0.866i)3-s + (0.500 − 0.866i)4-s + (0.5 − 0.866i)5-s + 1.73i·6-s + (1.5 + 2.59i)7-s − 3·8-s + (1.5 + 2.59i)9-s − 0.999·10-s + (1 + 1.73i)11-s + (−1.5 + 0.866i)12-s + (1 − 1.73i)13-s + (1.5 − 2.59i)14-s + (−1.5 + 0.866i)15-s + (0.500 + 0.866i)16-s + 4·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.866 − 0.499i)3-s + (0.250 − 0.433i)4-s + (0.223 − 0.387i)5-s + 0.707i·6-s + (0.566 + 0.981i)7-s − 1.06·8-s + (0.5 + 0.866i)9-s − 0.316·10-s + (0.301 + 0.522i)11-s + (−0.433 + 0.249i)12-s + (0.277 − 0.480i)13-s + (0.400 − 0.694i)14-s + (−0.387 + 0.223i)15-s + (0.125 + 0.216i)16-s + 0.970·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.471907 - 0.395977i\)
\(L(\frac12)\) \(\approx\) \(0.471907 - 0.395977i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.5 + 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (-7 + 12.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.65159655886958966745143223074, −14.59658692375947974589001105430, −12.78151316972881947096599673587, −12.00980902494966768987910317271, −11.00188679517165027781986960752, −9.837615013129431333840783247602, −8.322555898333893512031081570108, −6.35450648608688705638615978455, −5.24266860815982016464903221824, −1.84862926202098096068348380912, 4.00699985814989429442296444762, 6.05464717443858286278310923735, 7.12815326122585644153487374397, 8.643998743299345752265537982989, 10.31543031998370937400513058999, 11.24523102913156254918080906476, 12.43913824156724759237327963615, 14.13991424112448994889739419289, 15.24074878824178810826251383680, 16.52926656768029484927933544625

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