Properties

Label 2-45-45.2-c1-0-1
Degree $2$
Conductor $45$
Sign $0.944 + 0.329i$
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  + (0.0499 + 0.186i)2-s + (−0.806 − 1.53i)3-s + (1.69 − 0.981i)4-s + (−0.250 + 2.22i)5-s + (0.245 − 0.226i)6-s + (−2.35 + 0.632i)7-s + (0.540 + 0.540i)8-s + (−1.69 + 2.47i)9-s + (−0.426 + 0.0641i)10-s + (−2.14 − 1.23i)11-s + (−2.87 − 1.81i)12-s + (1.57 + 0.422i)13-s + (−0.235 − 0.407i)14-s + (3.60 − 1.40i)15-s + (1.88 − 3.27i)16-s + (−0.403 + 0.403i)17-s + ⋯
L(s)  = 1  + (0.0352 + 0.131i)2-s + (−0.465 − 0.885i)3-s + (0.849 − 0.490i)4-s + (−0.112 + 0.993i)5-s + (0.100 − 0.0925i)6-s + (−0.891 + 0.238i)7-s + (0.191 + 0.191i)8-s + (−0.566 + 0.823i)9-s + (−0.134 + 0.0202i)10-s + (−0.646 − 0.373i)11-s + (−0.829 − 0.523i)12-s + (0.436 + 0.117i)13-s + (−0.0629 − 0.108i)14-s + (0.931 − 0.363i)15-s + (0.472 − 0.818i)16-s + (−0.0979 + 0.0979i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.944 + 0.329i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1/2),\ 0.944 + 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.761866 - 0.129097i\)
\(L(\frac12)\) \(\approx\) \(0.761866 - 0.129097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.806 + 1.53i)T \)
5 \( 1 + (0.250 - 2.22i)T \)
good2 \( 1 + (-0.0499 - 0.186i)T + (-1.73 + i)T^{2} \)
7 \( 1 + (2.35 - 0.632i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (2.14 + 1.23i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.57 - 0.422i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.403 - 0.403i)T - 17iT^{2} \)
19 \( 1 - 4.28iT - 19T^{2} \)
23 \( 1 + (-1.82 + 6.82i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.20 + 5.55i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.97 + 3.41i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.171 + 0.171i)T + 37iT^{2} \)
41 \( 1 + (6.52 - 3.76i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.32 - 4.95i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.780 - 2.91i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.12 - 6.12i)T + 53iT^{2} \)
59 \( 1 + (-2.27 - 3.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.235 - 0.408i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.443 - 1.65i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.50iT - 71T^{2} \)
73 \( 1 + (-6.88 + 6.88i)T - 73iT^{2} \)
79 \( 1 + (-6.50 - 3.75i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.6 - 2.85i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 2.90T + 89T^{2} \)
97 \( 1 + (-1.41 + 0.379i)T + (84.0 - 48.5i)T^{2} \)
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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

−15.87773352407362402356476871236, −14.71123222179433061120675599798, −13.52263469483579180049893018241, −12.22180041206944105081009523937, −11.09476540279775538036667048541, −10.21211736757170213205916275412, −7.930583556439684391792978121854, −6.61845861056099245143074567744, −5.95380931451033480853255254751, −2.66325889852448419303599108902, 3.51034997305632338545401488175, 5.26536851551696707762357532343, 6.95944995477833459909175948171, 8.743281825479158358926770416023, 10.05062189178175412837507460436, 11.23598118171581382488023989688, 12.35295735437311642366007995859, 13.31346955019784067009470358295, 15.46940236013633195489576822759, 15.89745996437422580724254953366

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