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Properties

Label 2-45-45.4-c3-0-11
Degree $2$
Conductor $45$
Sign $0.830 + 0.556i$
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  + (4.35 − 2.51i)2-s + (1.99 + 4.79i)3-s + (8.61 − 14.9i)4-s + (−11.1 + 0.192i)5-s + (20.7 + 15.8i)6-s + (4.66 − 2.69i)7-s − 46.4i·8-s + (−19.0 + 19.1i)9-s + (−48.1 + 28.9i)10-s + (19.5 + 33.8i)11-s + (88.8 + 11.5i)12-s + (−75.0 − 43.3i)13-s + (13.5 − 23.4i)14-s + (−23.2 − 53.2i)15-s + (−47.6 − 82.4i)16-s − 15.4i·17-s + ⋯
L(s)  = 1  + (1.53 − 0.888i)2-s + (0.384 + 0.923i)3-s + (1.07 − 1.86i)4-s + (−0.999 + 0.0172i)5-s + (1.41 + 1.07i)6-s + (0.251 − 0.145i)7-s − 2.05i·8-s + (−0.704 + 0.709i)9-s + (−1.52 + 0.914i)10-s + (0.535 + 0.928i)11-s + (2.13 + 0.277i)12-s + (−1.60 − 0.924i)13-s + (0.258 − 0.446i)14-s + (−0.400 − 0.916i)15-s + (−0.744 − 1.28i)16-s − 0.219i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.830 + 0.556i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ 0.830 + 0.556i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.59496 - 0.788968i\)
\(L(\frac12)\) \(\approx\) \(2.59496 - 0.788968i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.99 - 4.79i)T \)
5 \( 1 + (11.1 - 0.192i)T \)
good2 \( 1 + (-4.35 + 2.51i)T + (4 - 6.92i)T^{2} \)
7 \( 1 + (-4.66 + 2.69i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-19.5 - 33.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (75.0 + 43.3i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 15.4iT - 4.91e3T^{2} \)
19 \( 1 - 26.8T + 6.85e3T^{2} \)
23 \( 1 + (-96.2 - 55.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-24.6 - 42.6i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-89.9 + 155. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 293. iT - 5.06e4T^{2} \)
41 \( 1 + (13.8 - 23.9i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-52.3 + 30.2i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (83.3 - 48.1i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 251. iT - 1.48e5T^{2} \)
59 \( 1 + (38.4 - 66.5i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (245. + 424. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (206. + 119. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 640.T + 3.57e5T^{2} \)
73 \( 1 - 769. iT - 3.89e5T^{2} \)
79 \( 1 + (-331. - 573. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (1.12e3 - 651. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 995.T + 7.04e5T^{2} \)
97 \( 1 + (-705. + 407. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−14.95072344784018866558434315076, −14.29184110133594223525984723104, −12.75689732574100006216173784514, −11.82690016737065022748480166857, −10.80591022111050369969109623231, −9.630032120241922369324601009806, −7.50204693514039156912776824945, −5.12886281333633752465212747318, −4.23789252230714024114847958819, −2.84802844515800209415220584959, 3.12431037874444961245536866161, 4.75085378641939389200703104494, 6.52106356024863797926084015036, 7.42246114718795584476077567316, 8.602219429499219226424819035848, 11.67856428699189348481848244131, 12.12014288374698204333021888428, 13.35191860063326394157112481142, 14.44145132373186059492110110775, 14.92881612383721004104659576757

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