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Properties

Label 2-45-45.4-c3-0-9
Degree $2$
Conductor $45$
Sign $0.999 + 0.00800i$
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  + (−0.680 + 0.392i)2-s + (3.65 − 3.69i)3-s + (−3.69 + 6.39i)4-s + (10.9 + 2.15i)5-s + (−1.03 + 3.94i)6-s + (18.1 − 10.4i)7-s − 12.0i·8-s + (−0.319 − 26.9i)9-s + (−8.30 + 2.83i)10-s + (29.6 + 51.4i)11-s + (10.1 + 36.9i)12-s + (−39.7 − 22.9i)13-s + (−8.21 + 14.2i)14-s + (48.0 − 32.6i)15-s + (−24.7 − 42.9i)16-s + 43.0i·17-s + ⋯
L(s)  = 1  + (−0.240 + 0.138i)2-s + (0.702 − 0.711i)3-s + (−0.461 + 0.799i)4-s + (0.981 + 0.193i)5-s + (−0.0702 + 0.268i)6-s + (0.977 − 0.564i)7-s − 0.533i·8-s + (−0.0118 − 0.999i)9-s + (−0.262 + 0.0897i)10-s + (0.813 + 1.40i)11-s + (0.244 + 0.890i)12-s + (−0.848 − 0.489i)13-s + (−0.156 + 0.271i)14-s + (0.827 − 0.562i)15-s + (−0.387 − 0.670i)16-s + 0.614i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00800i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.999 + 0.00800i$
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: \((2,\ 45,\ (\ :3/2),\ 0.999 + 0.00800i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.55520 - 0.00622855i\)
\(L(\frac12)\) \(\approx\) \(1.55520 - 0.00622855i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.65 + 3.69i)T \)
5 \( 1 + (-10.9 - 2.15i)T \)
good2 \( 1 + (0.680 - 0.392i)T + (4 - 6.92i)T^{2} \)
7 \( 1 + (-18.1 + 10.4i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-29.6 - 51.4i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (39.7 + 22.9i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 43.0iT - 4.91e3T^{2} \)
19 \( 1 + 140.T + 6.85e3T^{2} \)
23 \( 1 + (88.1 + 50.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-6.19 - 10.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (35.9 - 62.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 150. iT - 5.06e4T^{2} \)
41 \( 1 + (25.5 - 44.2i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-29.8 + 17.2i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-102. + 58.9i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 137. iT - 1.48e5T^{2} \)
59 \( 1 + (-248. + 429. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-123. - 214. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (715. + 413. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 260.T + 3.57e5T^{2} \)
73 \( 1 - 372. iT - 3.89e5T^{2} \)
79 \( 1 + (-238. - 412. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (271. - 156. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 817.T + 7.04e5T^{2} \)
97 \( 1 + (-815. + 470. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−14.83891572361977061770175453372, −14.28185004427875036458959100451, −12.99008467830804509913556994731, −12.27422640084753681890892291616, −10.23234776177659910746149294828, −9.005034030563049306623524884987, −7.82532517170647028929094419470, −6.74624993751117485195152219545, −4.31153785305181886858489795048, −2.03217801270584813019202199968, 2.03571555761637640087094423349, 4.63208020034592524355510110311, 5.88898592758370075919075449342, 8.512719153376590908331404135533, 9.160511683617933146818108068760, 10.28274730651826508953899267144, 11.46230864738788303721540859339, 13.53563248170796579905748783538, 14.31649431644751549024956325975, 14.93136718755001327402922332169

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