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Properties

Label 2-45-45.29-c4-0-8
Degree $2$
Conductor $45$
Sign $0.168 - 0.985i$
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  + (−3.64 + 6.30i)2-s + (4.56 − 7.75i)3-s + (−18.5 − 32.0i)4-s + (21.9 + 12.0i)5-s + (32.2 + 57.0i)6-s + (38.4 + 22.2i)7-s + 153.·8-s + (−39.3 − 70.8i)9-s + (−155. + 94.4i)10-s + (142. + 82.2i)11-s + (−333. − 2.80i)12-s + (−140. + 81.0i)13-s + (−280. + 161. i)14-s + (193. − 115. i)15-s + (−261. + 452. i)16-s + 79.6·17-s + ⋯
L(s)  = 1  + (−0.910 + 1.57i)2-s + (0.507 − 0.861i)3-s + (−1.15 − 2.00i)4-s + (0.876 + 0.481i)5-s + (0.896 + 1.58i)6-s + (0.784 + 0.453i)7-s + 2.39·8-s + (−0.485 − 0.874i)9-s + (−1.55 + 0.944i)10-s + (1.17 + 0.679i)11-s + (−2.31 − 0.0194i)12-s + (−0.830 + 0.479i)13-s + (−1.42 + 0.824i)14-s + (0.859 − 0.511i)15-s + (−1.02 + 1.76i)16-s + 0.275·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 - 0.985i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.168 - 0.985i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ 0.168 - 0.985i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.987466 + 0.832638i\)
\(L(\frac12)\) \(\approx\) \(0.987466 + 0.832638i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.56 + 7.75i)T \)
5 \( 1 + (-21.9 - 12.0i)T \)
good2 \( 1 + (3.64 - 6.30i)T + (-8 - 13.8i)T^{2} \)
7 \( 1 + (-38.4 - 22.2i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-142. - 82.2i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (140. - 81.0i)T + (1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 79.6T + 8.35e4T^{2} \)
19 \( 1 - 493.T + 1.30e5T^{2} \)
23 \( 1 + (-99.2 - 171. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-91.9 - 53.1i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (517. + 896. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 1.04e3iT - 1.87e6T^{2} \)
41 \( 1 + (-38.6 + 22.3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (2.98e3 + 1.72e3i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-418. + 724. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 307.T + 7.89e6T^{2} \)
59 \( 1 + (3.86e3 - 2.23e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-876. + 1.51e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (3.52e3 - 2.03e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 5.45e3iT - 2.54e7T^{2} \)
73 \( 1 - 486. iT - 2.83e7T^{2} \)
79 \( 1 + (-3.63e3 + 6.29e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-1.75e3 + 3.03e3i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 5.21e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.51e3 - 871. i)T + (4.42e7 + 7.66e7i)T^{2} \)
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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

−15.01817316919299471980947262911, −14.54092791222606041706531532026, −13.71994481301948075912511351302, −11.84252222489808121079567247243, −9.715467962296888563964945228650, −9.016079942362914780117430445403, −7.56600167115423078721520040763, −6.77378860080874726817883516766, −5.45699747194281236165354475354, −1.64923606365922534483694633676, 1.36640973858027241258351311623, 3.17434490740938736489930917015, 4.83306347908231935085291636848, 8.065612669370485161371983135792, 9.159238188798647511649249703869, 9.910690116287928765163755891190, 10.95268605847799467974672738631, 12.06676169921138875854361639639, 13.57097129605656617372997773069, 14.43859205567046423625375067742

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