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Properties

Label 2-45-9.7-c3-0-4
Degree $2$
Conductor $45$
Sign $0.909 - 0.415i$
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  + (1.09 − 1.90i)2-s + (0.206 + 5.19i)3-s + (1.59 + 2.75i)4-s + (2.5 + 4.33i)5-s + (10.0 + 5.30i)6-s + (−1.38 + 2.39i)7-s + 24.5·8-s + (−26.9 + 2.14i)9-s + 10.9·10-s + (26.3 − 45.6i)11-s + (−13.9 + 8.83i)12-s + (−10.2 − 17.7i)13-s + (3.03 + 5.25i)14-s + (−21.9 + 13.8i)15-s + (14.1 − 24.5i)16-s + 3.66·17-s + ⋯
L(s)  = 1  + (0.387 − 0.671i)2-s + (0.0396 + 0.999i)3-s + (0.199 + 0.344i)4-s + (0.223 + 0.387i)5-s + (0.686 + 0.360i)6-s + (−0.0746 + 0.129i)7-s + 1.08·8-s + (−0.996 + 0.0792i)9-s + 0.346·10-s + (0.721 − 1.25i)11-s + (−0.336 + 0.212i)12-s + (−0.218 − 0.377i)13-s + (0.0579 + 0.100i)14-s + (−0.378 + 0.238i)15-s + (0.221 − 0.384i)16-s + 0.0522·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.909 - 0.415i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ 0.909 - 0.415i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.71810 + 0.373779i\)
\(L(\frac12)\) \(\approx\) \(1.71810 + 0.373779i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.206 - 5.19i)T \)
5 \( 1 + (-2.5 - 4.33i)T \)
good2 \( 1 + (-1.09 + 1.90i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (1.38 - 2.39i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-26.3 + 45.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (10.2 + 17.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 3.66T + 4.91e3T^{2} \)
19 \( 1 + 95.6T + 6.85e3T^{2} \)
23 \( 1 + (-44.9 - 77.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-113. + 197. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (139. + 241. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 273.T + 5.06e4T^{2} \)
41 \( 1 + (32.4 + 56.1i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (209. - 362. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (69.3 - 120. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 197.T + 1.48e5T^{2} \)
59 \( 1 + (370. + 641. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (244. - 423. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-205. - 356. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 310.T + 3.57e5T^{2} \)
73 \( 1 + 51.0T + 3.89e5T^{2} \)
79 \( 1 + (603. - 1.04e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (452. - 783. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 663.T + 7.04e5T^{2} \)
97 \( 1 + (-362. + 628. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−15.35876474258603730487113294729, −14.21573323287484437585498771984, −13.09419552675260514150801298924, −11.54574265666053132958820189518, −10.93143290652637082849829971485, −9.612340543406932584995769846780, −8.128459750527649550931073912078, −6.06396779030131570308690092959, −4.16526031730662197024544977288, −2.86768336609205364704136084944, 1.72566718424647857131171102457, 4.81803938443778568158602239684, 6.43465433743900586591734761852, 7.20449222918766719212337161386, 8.847603986805785603106411855655, 10.50790362395910676652623712666, 12.11053118037940753693702868441, 13.04831087746018004945942824464, 14.29052028288602535521028124400, 14.92371691741077839145703605521

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