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Properties

Label 2-45-9.4-c1-0-0
Degree $2$
Conductor $45$
Sign $0.999 + 0.0173i$
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  + (−0.285 − 0.495i)2-s + (0.285 + 1.70i)3-s + (0.836 − 1.44i)4-s + (−0.5 + 0.866i)5-s + (0.764 − 0.630i)6-s + (−0.714 − 1.23i)7-s − 2.10·8-s + (−2.83 + 0.977i)9-s + 0.571·10-s + (−1.33 − 2.31i)11-s + (2.71 + 1.01i)12-s + (−2.33 + 4.04i)13-s + (−0.408 + 0.707i)14-s + (−1.62 − 0.606i)15-s + (−1.07 − 1.85i)16-s + 2.67·17-s + ⋯
L(s)  = 1  + (−0.202 − 0.350i)2-s + (0.165 + 0.986i)3-s + (0.418 − 0.724i)4-s + (−0.223 + 0.387i)5-s + (0.312 − 0.257i)6-s + (−0.269 − 0.467i)7-s − 0.742·8-s + (−0.945 + 0.325i)9-s + 0.180·10-s + (−0.402 − 0.697i)11-s + (0.783 + 0.292i)12-s + (−0.648 + 1.12i)13-s + (−0.109 + 0.189i)14-s + (−0.418 − 0.156i)15-s + (−0.267 − 0.464i)16-s + 0.648·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.999 + 0.0173i$
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: \((2,\ 45,\ (\ :1/2),\ 0.999 + 0.0173i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.759796 - 0.00657468i\)
\(L(\frac12)\) \(\approx\) \(0.759796 - 0.00657468i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.285 - 1.70i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.285 + 0.495i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (0.714 + 1.23i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.33 + 2.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.33 - 4.04i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.67T + 17T^{2} \)
19 \( 1 - 4.67T + 19T^{2} \)
23 \( 1 + (-2.95 + 5.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.74 - 8.21i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.48 - 6.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.81T + 37T^{2} \)
41 \( 1 + (-0.735 + 1.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.235 + 0.408i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.47 + 6.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.14T + 53T^{2} \)
59 \( 1 + (-0.571 + 0.990i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.26 - 2.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.29 + 5.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + 1.71T + 73T^{2} \)
79 \( 1 + (-0.143 - 0.249i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.14 - 3.71i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (3.91 + 6.78i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.99891136939233648747677686354, −14.65782820857765487869413824386, −14.05913989684097082650737061262, −11.95441576373610528950077135057, −10.83542569523987005041963807753, −10.10105042524175183232538873076, −8.893951282230359066206896472447, −6.90406840563948206761509656245, −5.16955999748238025204935450543, −3.14319303985762875120891141004, 2.88630941288728393362215202062, 5.70657316268924195173900432208, 7.38995516787053244561912536982, 7.997343931752726173805196531392, 9.517499571512560640742638575240, 11.64981260226710242967313892980, 12.43425909057846734327392994612, 13.26539609170015479572141596592, 14.97130758299893278305717536386, 15.86743016413490047852388663725

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