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Properties

Label 2-45-45.34-c3-0-12
Degree $2$
Conductor $45$
Sign $-0.849 + 0.528i$
Analytic cond. $2.65508$
Root an. cond. $1.62944$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.67 − 1.54i)2-s + (2.05 − 4.77i)3-s + (0.766 + 1.32i)4-s + (11.1 + 0.287i)5-s + (−12.8 + 9.58i)6-s + (−27.1 − 15.6i)7-s + 19.9i·8-s + (−18.5 − 19.6i)9-s + (−29.4 − 18.0i)10-s + (9.59 − 16.6i)11-s + (7.90 − 0.926i)12-s + (−18.1 + 10.4i)13-s + (48.3 + 83.7i)14-s + (24.3 − 52.7i)15-s + (36.9 − 64.0i)16-s + 6.19i·17-s + ⋯
L(s)  = 1  + (−0.945 − 0.545i)2-s + (0.395 − 0.918i)3-s + (0.0957 + 0.165i)4-s + (0.999 + 0.0257i)5-s + (−0.875 + 0.652i)6-s + (−1.46 − 0.845i)7-s + 0.882i·8-s + (−0.686 − 0.726i)9-s + (−0.930 − 0.569i)10-s + (0.263 − 0.455i)11-s + (0.190 − 0.0222i)12-s + (−0.387 + 0.223i)13-s + (0.922 + 1.59i)14-s + (0.419 − 0.907i)15-s + (0.577 − 1.00i)16-s + 0.0883i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.528i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.849 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.849 + 0.528i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ -0.849 + 0.528i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.227110 - 0.794738i\)
\(L(\frac12)\) \(\approx\) \(0.227110 - 0.794738i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.05 + 4.77i)T \)
5 \( 1 + (-11.1 - 0.287i)T \)
good2 \( 1 + (2.67 + 1.54i)T + (4 + 6.92i)T^{2} \)
7 \( 1 + (27.1 + 15.6i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-9.59 + 16.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (18.1 - 10.4i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 6.19iT - 4.91e3T^{2} \)
19 \( 1 - 96.6T + 6.85e3T^{2} \)
23 \( 1 + (-141. + 81.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-3.82 + 6.62i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (112. + 194. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 155. iT - 5.06e4T^{2} \)
41 \( 1 + (-157. - 272. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-166. - 96.3i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-275. - 159. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 277. iT - 1.48e5T^{2} \)
59 \( 1 + (214. + 371. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-44.9 + 77.8i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (505. - 291. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 132.T + 3.57e5T^{2} \)
73 \( 1 - 259. iT - 3.89e5T^{2} \)
79 \( 1 + (-62.3 + 107. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (31.8 + 18.3i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 333.T + 7.04e5T^{2} \)
97 \( 1 + (891. + 514. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−14.45653965859116430138123488855, −13.60328433047673822244409758190, −12.66419844978440628416776456626, −11.01730598986108013711505767349, −9.699475470042992663041852641240, −9.104663748435281816862082839767, −7.33179735656126265567324944191, −6.02625424134904360803006166061, −2.81156089160981092838902913580, −0.886122554349239319794878113298, 3.12639077734205265955055104028, 5.52387277453851448110133452965, 7.07443278273477624350570233368, 9.013083462337706136887083229209, 9.391830201005108394413276729377, 10.29738109783058531756538622995, 12.45878298394585124025360366992, 13.61380495883000748162390465739, 15.12156407135268816380912310750, 15.98262318775003978807461014125

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