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Properties

Label 2-45-45.29-c4-0-5
Degree $2$
Conductor $45$
Sign $-0.174 - 0.984i$
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  + (0.316 − 0.548i)2-s + (4.90 + 7.54i)3-s + (7.79 + 13.5i)4-s + (−18.0 + 17.3i)5-s + (5.68 − 0.303i)6-s + (−73.2 − 42.2i)7-s + 20.0·8-s + (−32.8 + 74.0i)9-s + (3.80 + 15.3i)10-s + (148. + 85.7i)11-s + (−63.6 + 125. i)12-s + (127. − 73.4i)13-s + (−46.3 + 26.7i)14-s + (−219. − 50.7i)15-s + (−118. + 205. i)16-s + 273.·17-s + ⋯
L(s)  = 1  + (0.0791 − 0.137i)2-s + (0.545 + 0.838i)3-s + (0.487 + 0.844i)4-s + (−0.720 + 0.693i)5-s + (0.158 − 0.00843i)6-s + (−1.49 − 0.862i)7-s + 0.312·8-s + (−0.404 + 0.914i)9-s + (0.0380 + 0.153i)10-s + (1.22 + 0.708i)11-s + (−0.441 + 0.869i)12-s + (0.752 − 0.434i)13-s + (−0.236 + 0.136i)14-s + (−0.974 − 0.225i)15-s + (−0.462 + 0.801i)16-s + 0.946·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.174 - 0.984i$
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: \((2,\ 45,\ (\ :2),\ -0.174 - 0.984i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.04683 + 1.24821i\)
\(L(\frac12)\) \(\approx\) \(1.04683 + 1.24821i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.90 - 7.54i)T \)
5 \( 1 + (18.0 - 17.3i)T \)
good2 \( 1 + (-0.316 + 0.548i)T + (-8 - 13.8i)T^{2} \)
7 \( 1 + (73.2 + 42.2i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-148. - 85.7i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-127. + 73.4i)T + (1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 273.T + 8.35e4T^{2} \)
19 \( 1 - 229.T + 1.30e5T^{2} \)
23 \( 1 + (-194. - 337. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-306. - 177. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (310. + 538. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 730. iT - 1.87e6T^{2} \)
41 \( 1 + (498. - 288. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (783. + 452. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-396. + 686. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 3.46e3T + 7.89e6T^{2} \)
59 \( 1 + (-2.43e3 + 1.40e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-2.66e3 + 4.60e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-147. + 85.4i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 2.79e3iT - 2.54e7T^{2} \)
73 \( 1 - 5.86e3iT - 2.83e7T^{2} \)
79 \( 1 + (-1.46e3 + 2.54e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-3.44e3 + 5.96e3i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 1.03e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.21e4 - 7.02e3i)T + (4.42e7 + 7.66e7i)T^{2} \)
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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

−15.61794715881663922707822374902, −14.34163116757194431385477773594, −13.09550358296227681721124564383, −11.77814693785492886966061429911, −10.57822042680100278182585543211, −9.470417754191788861660721223948, −7.78623756266002742584559072080, −6.72111731541671443210470108097, −3.80905018652513994502485838088, −3.30566654226296180043432886241, 1.04950169019918407090870396814, 3.31447363041868076543394174942, 5.90798882945445278620204530781, 6.84695818643641001643069133201, 8.627922680851681327632711403596, 9.529733626328466600576029916223, 11.58512273040454289564917644055, 12.37757551203950746928664594486, 13.62914438874996002316424128437, 14.75751812681774766205853776121

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