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Properties

Label 2-45-9.5-c4-0-0
Degree $2$
Conductor $45$
Sign $-0.474 + 0.880i$
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  + (−2.15 + 1.24i)2-s + (−6.81 + 5.88i)3-s + (−4.91 + 8.51i)4-s + (9.68 + 5.59i)5-s + (7.34 − 21.1i)6-s + (−15.7 − 27.3i)7-s − 64.1i·8-s + (11.7 − 80.1i)9-s − 27.7·10-s + (−182. + 105. i)11-s + (−16.6 − 86.9i)12-s + (143. − 248. i)13-s + (67.8 + 39.1i)14-s + (−98.8 + 18.8i)15-s + (0.981 + 1.69i)16-s + 195. i·17-s + ⋯
L(s)  = 1  + (−0.537 + 0.310i)2-s + (−0.756 + 0.653i)3-s + (−0.307 + 0.532i)4-s + (0.387 + 0.223i)5-s + (0.203 − 0.586i)6-s + (−0.322 − 0.557i)7-s − 1.00i·8-s + (0.145 − 0.989i)9-s − 0.277·10-s + (−1.51 + 0.872i)11-s + (−0.115 − 0.603i)12-s + (0.849 − 1.47i)13-s + (0.346 + 0.199i)14-s + (−0.439 + 0.0839i)15-s + (0.00383 + 0.00664i)16-s + 0.675i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.474 + 0.880i$
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: \((2,\ 45,\ (\ :2),\ -0.474 + 0.880i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.00858492 - 0.0143870i\)
\(L(\frac12)\) \(\approx\) \(0.00858492 - 0.0143870i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (6.81 - 5.88i)T \)
5 \( 1 + (-9.68 - 5.59i)T \)
good2 \( 1 + (2.15 - 1.24i)T + (8 - 13.8i)T^{2} \)
7 \( 1 + (15.7 + 27.3i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (182. - 105. i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-143. + 248. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 195. iT - 8.35e4T^{2} \)
19 \( 1 + 139.T + 1.30e5T^{2} \)
23 \( 1 + (409. + 236. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (1.00e3 - 583. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (160. - 277. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 2.15e3T + 1.87e6T^{2} \)
41 \( 1 + (520. + 300. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (1.38e3 + 2.40e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (692. - 399. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 78.0iT - 7.89e6T^{2} \)
59 \( 1 + (-893. - 515. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-2.58e3 - 4.47e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (2.03e3 - 3.52e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 2.63e3iT - 2.54e7T^{2} \)
73 \( 1 - 5.27e3T + 2.83e7T^{2} \)
79 \( 1 + (-2.45e3 - 4.25e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (3.13e3 - 1.80e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 8.33e3iT - 6.27e7T^{2} \)
97 \( 1 + (319. + 554. i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−15.98113358498211155383011089296, −15.19759891261293041798814345702, −13.24627310676244647708185120604, −12.53869959685051824526311918596, −10.50394372165613554128817853984, −10.16098404470689902000971516972, −8.476513799183576276850660385430, −7.07320347023077161477329684540, −5.44398718881288726395449218861, −3.66109747396982852106610617360, 0.01340694908063972504992157354, 1.94402081017444169977963088773, 5.22366253134904099759825459247, 6.22409972041481523436305596715, 8.180894666653978478557618894511, 9.447497872779651486882792867022, 10.77711915094935696338733017873, 11.66773791898904379158171014582, 13.22755412231224263010690135909, 13.90081350316327758619733029631

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