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Properties

Label 2-45-9.5-c4-0-15
Degree $2$
Conductor $45$
Sign $-0.470 + 0.882i$
Analytic cond. $4.65164$
Root an. cond. $2.15676$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.93 − 3.42i)2-s + (−5.90 − 6.79i)3-s + (15.5 − 26.8i)4-s + (−9.68 − 5.59i)5-s + (−58.3 − 20.1i)6-s + (12.5 + 21.7i)7-s − 102. i·8-s + (−11.3 + 80.2i)9-s − 76.6·10-s + (112. − 64.6i)11-s + (−273. + 53.1i)12-s + (143. − 249. i)13-s + (149. + 86.0i)14-s + (19.1 + 98.7i)15-s + (−104. − 181. i)16-s + 400. i·17-s + ⋯
L(s)  = 1  + (1.48 − 0.856i)2-s + (−0.655 − 0.755i)3-s + (0.968 − 1.67i)4-s + (−0.387 − 0.223i)5-s + (−1.62 − 0.558i)6-s + (0.256 + 0.443i)7-s − 1.60i·8-s + (−0.140 + 0.990i)9-s − 0.766·10-s + (0.925 − 0.534i)11-s + (−1.90 + 0.368i)12-s + (0.850 − 1.47i)13-s + (0.760 + 0.439i)14-s + (0.0851 + 0.439i)15-s + (−0.408 − 0.707i)16-s + 1.38i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.470 + 0.882i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ -0.470 + 0.882i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.34211 - 2.23619i\)
\(L(\frac12)\) \(\approx\) \(1.34211 - 2.23619i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.90 + 6.79i)T \)
5 \( 1 + (9.68 + 5.59i)T \)
good2 \( 1 + (-5.93 + 3.42i)T + (8 - 13.8i)T^{2} \)
7 \( 1 + (-12.5 - 21.7i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-112. + 64.6i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-143. + 249. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 400. iT - 8.35e4T^{2} \)
19 \( 1 + 74.2T + 1.30e5T^{2} \)
23 \( 1 + (-143. - 83.0i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (972. - 561. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (529. - 916. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 1.50e3T + 1.87e6T^{2} \)
41 \( 1 + (1.91e3 + 1.10e3i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-665. - 1.15e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (3.02e3 - 1.74e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 1.03e3iT - 7.89e6T^{2} \)
59 \( 1 + (521. + 301. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (2.86e3 + 4.96e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (23.5 - 40.7i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 2.80e3iT - 2.54e7T^{2} \)
73 \( 1 - 5.27e3T + 2.83e7T^{2} \)
79 \( 1 + (-3.82e3 - 6.62e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (6.44e3 - 3.72e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 4.96e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.53e3 - 2.65e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−14.39799321421441392794399932424, −13.04527610120890252063355029555, −12.59332407955687392176829838378, −11.40282745909760791790802121507, −10.75340236243232167639824253431, −8.296558381461939812910088355904, −6.27224497593503019354895607081, −5.30337955156360956735935914391, −3.53949969848946352910314782733, −1.43998910338996529126417935110, 3.83588159017607745425707372118, 4.63777498764390983191841694375, 6.23687053279486765602239734112, 7.22198237910722459580874275308, 9.351675679647514190278175461675, 11.33245737709201306790598068518, 11.86577635868730508991786297107, 13.46292433505612617327504944185, 14.50734908680257172988531702127, 15.27087543868445092218172557652

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